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CONCRETE   GEOMETRY 


FOR  BEGINNERS 


BY 


A.   R.   HORNBROOK,   A.M. 

Teacher  of  Mathematics  en  High  School,  Evansville,  Ind. 


3>»ic 


NEW  YORK.:. CINCINNATI.:. CHICAGO 

AMERICAN    BOOK   COMPANY 


COPTBIOHT,   1895,   BT 

AMERICAN   BOOK  COMPANY. 


A.   R.   H.   COXCRKTB  QEOH. 
w.  p.  3 


UNIVER^ 

Of 


PREFACE. 


"If  you  will  put  that  into  figures  for  me,  perhaps  I  can  under- 
stand it,"  was  said  to  the  author  of  this  book  by  a  pupil,  who,  having 
tried  in  vain  to  gi*asp  a  general  proposition  in  demonstrative  geome- 
try, sought  a  kind  of  assistance  which  he  had  found  to  be  useful. 
In  his  demand  for  figures  (by  which  he  meant  numbers)  he  was 
expressing  the  universal  demand  of  the  learning  mind  for  the  concrete 
and  the  particular,  as  stepping  stones  to  the  abstract  and  the  general. 

Much  of  the  material  of  this  book  was  prepared  to  meet  this 
demand  on  the  part  of  the  author's  own  pupils  during  several  years 
of  teaching,  and  all  of  it  has  been  subjected  to  the  test  of  the  school- 
room under  plans  which  encourage  on  the  part  of  the  pupils  the 
freest  disclosures  of  the  mental  processes  induced  by  the  work. 

The  aim  of  the  work  is  to  awaken  gradually,  by  simple  and 
natm-al  methods,  the  mathematical  consciousness  of  the  child  and 
to  guide  his  perceptions  in  such  a  way  as  to  lead  him  to  lay  a  firm 
foundation  for  demonstrative  geometry  by  means  of  his  own  observa- 
tions and  inventions.  The  recognition  of  different  geometric  forms 
and  of  their  numerical  relations,  and  the  practicing  of  the  geometric 
imagination,  constitute  for  children  a  useful  and  delightful  exercise, 
which  this  book  is  intended  to  promote. 

The  author  has  selected  from  standard  mathematical  works  the 
statements  of  some  important  facts  and  principles  that  are  capable  of 
concrete  demonstration  from  the  standpoint  of  the  child,  and  has 
presented  them  in  various  ways  and  in  different  relations.  Emphasis 
is  placed  upon  the  number  relations  of  geometric  forms,  as  a  means 
of  arousing  those  definite  and  exact  ideas  of  form  that  are  induced 
by  actual  measurements  and  computations. 

As  a  step  towards  the  correlation  of  studies  which  is  acknowl- 
edged to  promote  economy  in  educational  effort,  the  algebraic 
equation   in  its  simplest  forms  and  uses  has  been   introduced,   as 

3 

183664 


4  PREFACE. 

a  convenient  instrument  for  determining  values  in  connection  with 
geometric  forms.  Teachers  who  have  used  the  algebraic  equation 
in  arithmetical  or  geometrical  work  are  agi'eed  that  its  use  not  only 
facilitates  the  work  in  hand,  but  that  it  tends  to  establish  the  habit 
of  readily  symbolizing  the  unknown,  which  is  so  valuable  in  mathe- 
matical work. 

The  author  has  tried,  at  the  expense  sometimes  of  completeness 
in  arrangement,  to  avoid  the  pedagogical  blunder  of  putting  similars 
in  proximity.  Supplement  and  complement,  rhomboid  and  rhombus, 
trapezoid  and  trapezium,  are  separated  in  presentation,  with  the 
hope  that  the  first  one  presented  will  have  made  an  ineffaceable 
impression  of  itself  upon  the  miud  of  the  pupil  before  the  other 
is  received. 

Although  designed  especially  for  use  in  grammar  grades  in  ac- 
cordance with  the  recommendations  of  the  Committee  of  Ten  and 
with  the  practice  of  many  of  our  foremost  schools,  this  book  will 
be  found  useful  for  supplementary  work  for  beginners  in  demonstra- 
tive geometry  or  for  a  rapid  preliminary  drill  for  such  students. 
Too  often  the  pupil  who  can  recite  glibly  demonstration  after 
demonstration  of  geometric  principles  is  unable  to  make  the  sim- 
plest applications  of  them,  a  fact  which  shows  that  his  work  in 
geometry  is  not  accomplishing  its  object  in  developing  his  mathe- 
matical powers.  The  author  trusts  that  the  material  of  this  book 
will  be  found  useful  to  teachers  in  helping  them  to  establish  in 
the  minds  of  learners  the  habit  of  making  applications  of  mathe- 
matical truths. 

Grateful  acknowledgments  for  assistance  in  revising  proof  sheets 
and  for  valuable  suggestions  upon  the  work  are  due  to  Prof.  O.  L. 
Kelso  of  the  State  Normal  School,  Terre  Haute,  Ind.  and  to  Prof. 
S.  C.  Davisson  of  the  Department  of  Mathematics  of  the  University 
of  Indiana. 

Further  suggestions  and  criticisms  from  teachers  will  be  gladly 
received.  This  little  book  is  sent  out  with  the  hope  that  its  use  may 
be  as  enjoyable  to  other  teachers  as  the  writing  of  it  and  its  applica- 
tions in  the  schoolroom  have  been  to  the  author. 


SUGGESTIONS  TO   TEACHERS. 


The  general  method  of  the  pupil's  work  in  this  book  is  that 
of  constructing  and  inspecting  geometric  forms  and  of  report- 
ing in  the  language  of  mathematics  the  results  of  the  inspec- 
tion.    His  success  depends  very  largely  upon  the  teacher. 

Experienced  teachers  do  not  need  to  be  reminded  that  great 
care  is  necessary  in  order  that  each  pupil  shall  construct  care- 
fully, inspect  thoroughly,  and  report  exactly. 

The  following  suggestions  may  be  found  helpful : 

Do  not  attempt  to  teach  without  models  the  measurements 
of  cubes  or  other  solids.  While  those  measurements  are  very 
simple  when  made  upon  actual  solids,  it  is  beyond  the  power 
of  the  untrained  geometric  imagination  to  furnish  them.  Tlie 
object  of  this  work  being  to  secure  clear  thinking,  the  pupil 
should  not  be  allowed  to  form  and  use  false  or  imperfect 
mental  images  of  geometric  forms. 

Rulers,  protractors,  and  dividers,  or  a  substitute  for  them, 
are  an  absolute  necessity  in  this  work.  The  units  of  metric 
measurement  are  to  be  presented  objectively.  Very  cheap 
rulers  showing  decimeter,  centimeter,  and  millimeter  can  he 
obtained.  A  meter-stick  marked  like  a  yard-stick  ^nth  din- 
sions  should  be  made  and  used.     A  square  meter  and  its  sub- 


6  SUGGESTIONS    TO    TEACHERS, 

divisions  marked  on  the  floor  or  wall  of  the  schoolroom  will 
furnish  the  standard  of  thinking  when  the  metric  units  of  a 
problem  are  squares.  A  convenient  form  of  protractor  will  be 
found  at  the  end  of  this  work,  which  may  be  detached  without 
injuring  the  book.  It  will  serve  as  a  general  model  in  the 
construction  of  protractors. 

All  solutions  that  use  superposition  as  in  G.C.M.  and  L.C.M. 
should  be  actually  performed  by  cutting  out  and  superposing 
until  each  child  knows  perfectly  the  significance  of  the  opera- 
tions—  but  no  longer.  The  skill  of  the  teacher  in  recognizing 
the  moment  when  the  pupil  has  gained  clear  and  correct  con- 
cepts of  that  which  he  illustrates,  and  in  stopping  the  pro- 
cesses of  illustration  before  they  degenerate  into  wearisome 
and  time-eating  formalities,  is  like  that  of  the  physician  who 
wisely  adjusts  his  treatment  of  a  case  to  its  varying  condi- 
tions. 

The  exercise  of  the  geometric  imagination  should  be  en- 
couraged by  allowing  pupils  to  substitute  mental  images  for 
physical  solids  as  soon  as  the  teacher  is  absolutely  sure  that 
those  images  are  accurate  and  complete.  It  is  evident  that 
the  establishment  of  sympathetic  relations  between  teacher 
and  pupil  is  a  very  direct  means  of  gaining  this  assurance  and 
at  the  same  time  of  facilitating  the  progress  of  the  learner. 


CONTENTS. 


OHAP. 

I.  Lines  and  Angles 
II.   Circles 
III.   Arcs  and  Angles 
rV.   Cumulative  Review    No.  1 
V.   Rectangles 
VI.   Triangles  and  Lines 
VII.   Cumulative  Review    No, 
VIII.   Quadrilaterals 
IX.   Ratio  and  Proportion 
X.   Cumulative  Review    No 
XI.   Polygons 
XII.   Circles  and  Lines     . 

XIII.  Cumulative  Review    No 

XIV.  Squares  and  Cubes   . 

XV.   Cumulative  Review    No.  5 

7 


9 

20 

27 

43 

52 

65 

89 

97 

113 

125 

132 

146 

160 

170 

192 


CHAPTER    I. 


LINES  AND  ANGLES. 


CuBVED  Line. 


Double  Cukyk. 


Broken  Lines. 


1.  What  capital  letter  in  print  is  formed  by  a  straight 
line  and  two  curves  ? 

2.  Which  of  the  capitals  is  formed  by  a  double  curve? 

3.  Draw  a  broken  line  consisting  of  two  straight  lines. 

4.  Name  two  capital  letters  each  of  which  is  formed 
by  a  broken  line  consisting  of  two  straight  lines. 

5.  Name  two  capitals  each  of  which  is  formed  by  a 
brojien  line  consisting  of  three  straight  lines. 

6.  Name  two  capitals  each  of  which  is  formed  by  a 
broken  line  consisting  of  four  straight  lines. 

7.  What  is  a  broken  line  ? 

8.  Mark  tAVO  points  A  and  B,  and  connect  them  by 
(1)  a  straight  line,  (2)  a  broken  line,  (3)  a  curve,  and 
(4)  a  double  curve.  Which  is  the  shortest  of  the  con- 
necting lines  ? 


10  LIXES  AND  ANGLES. 

9.    Read  the  following  geometric  principle  and  see  if 
it  is  true  : 

Principle  1. — A  straight  line  is  tlie  slwrtest  dis- 
tance between  two  points. 

10.  Mark  two  points  C  and  D  on  a  piece  of  paper,  and 
find  how  many  straight  lines  can  be  drawn  from  one  to 
the  other. 

11.  How  many  broken  lines  can  be  drawn  between  C 
andD? 

12.  How  many  curved  lines  can  be  drawn  between  C 
andD? 

13.  Can  you  show  the  truth  of  the  following  state- 
ment? 

Principle  2.  —  Between  two  points  only  one  straight 
line  can  be  drawn. 

14.  If  you  mark  two  points  on  the  surface  of  a  ball,  and 
run  a  straight  line  joining  the  points,  will  the  straight  line 
lie  on  the  surface  of  the  ball  ?     Illustrate  your  answer. 

15.  If  you  mark  two  points  on  a  flat  surface,  and  con- 
nect them  by  a  straight  line,  will  the  line  lie  wholly  in 
the  surface  ? 

Note.  —  A  surface  such  that  if  any  two  of  its  points  are  joined  hy 
a  straight  line,  tlie  line  will  lie  wholly  in  the  surface,  is  called  a  Plane 
Surface  or  Plane. 

IG.  If  the  waves  of  the  ocean  were  stilled,  would  its 
surface  be  a  plane  ? 

17.  Can  a  ship  on  the  ocean  sail  in  a  straight  line  ? 

18.  Can  you  draw  a  straight  line  on  the  surface  of  a 
ball? 


LINES  AND  ANGLES.  H 

19.  Can  you  draw  a  curved  line  on  a  plane  surface  ? 
Illustrate. 

20.  Make  an  angle  by  drawing  two  lines  that  meet. 
Note.  —  The  word  "  line  "  is  used  to  mean  straight  line. 

21.  Find  a  square  corner  on  the  floor.  On  the  wall. 
On  the  ceiling.     On  a  page  of  your  book.    On  your  ruler. 

22.  Two  lines  which  meet  so  as  to  make  a  square 
corner  form  a  right  angle.  Make  a  right  angle  by  draw- 
ing  along  the  edges  of  an  object  which  has  a  square 
corner. 

23.  Draw  an  acute  angle. 

Note. — An  Acute  Angle  is  an  angle  less  than  a  right  angle. 

24.  Draw  an  obtuse  angle. 

Note.  —  An  Obtuse  Angle  is  one  that  is  greater  than  a  right  angle. 

25.  Place  the  points  of  two  pencils  together  in  such  a 
way  as  to  make  a  right  angle.  Keeping  the  points 
together,  separate  the  ends  of  the  pencils  more.  What 
kind  of  an  angle  is  formed  by  the  pencils  ?  If  the  ends 
were  brought  nearer  than  when  the  pencils  formed  a  right 
angle,  what  kind  of  an  angle  would  be  formed  ? 

26.  How  many  angles  are  formed  by  the  lines  of  the 
letter  W,  and  of  what  kind  are  they  ? 

27.  Give  the  number  and  kinds  of  angles  found  in  the 
letter  N,  in  T,  in  X,  in  Y,  in  Z. 

28.  Give  the  number  of  each  kind  of  angle  found  in 
the  letters  of  the  word  AWAKE.     In  your  name. 

29.  Give  the  number  of  each  kind  of  angle  found  in 
the  letters  of  the  phrase  THE  NEXT  TIME. 


12  LINES  AND  ANGLES. 

30.  Give  the  number  of  each  kind  of  angle  found  in 
the  sentence  YET  THEY  WILL  TAKE  THE  TAX  AWAY. 

31.  Make  two  right  angles  with  two  lines. 

Note.  —  When  one  line  meets  another  bo  as  to  make  two  adjacent 
equal  angles,  the  lines  are  said  to  l>e  Perpendicular  to  each  other  and 
the  angles  are  called  Right  Angles. 

32.  Make  four  right  angles  with  two  lines. 

33.  Make  one  right  angle  with  two  lines.  Mark  its 
vertex  A^  and  the  ends  of  the  lines,  B  and  (7. 

XoTK.  —  The  point  where  two  lines  meet  is  called  the  vertex  of 
the  angle.     In  reading  an  angle,  the  letter  at  the  vertex  should  be 

placed  between  the  other  two;  thus,  the  angle  A  \^       \b  read,  the 
angle  CAD  or  DAC, 

84.  Name  the  two  adjacent  angles  at 
the  point  B,  What  line  belongs  to  both 
of  those  angles  ? 


35.    Draw  a  line  AB  and  place  an  angle 
DEF  ^o  that  ita  vertex  shall  be  somewhere  on  the  line 
AB^  and  no  other  iK>int  of  the  angle  shall  touch  that  line. 

86.    With  three  lines  make  three  angles. 

37.  Draw  three  lines  inclosing  a  surface.     How  many 
angles  are  thus  formed  ? 

Note.  —  A  figure  bounded  by  three  straight  lines  is  called  a 
Triangle. 

38.  Draw  a  plane  figure  bounded  by  three  lines  and 
^v^ite  its  name. 

Note.  —  When  an  inclosed  surface  is  a  plane  it  is  called  a  Plane 
Figure. 

39.  Can  you  draw  a  plane  figure  on  the  surface  of  a 
ball? 


LINES  AND  ANGLES.  18 

40.  Can  you  draw  a  plane  figure  on  the  surface  of  a 
stovepipe  ? 

41.  Can  you  draw  a  straight  line  on  the  surface  of  a 
stovepipe  ? 

42.  How  many  lines  form  the  perimeter  of  a  triangle  ? 

Note.  —  The  boundary  line  of  an  inclosed  surface  is  called  its 
Perimeter. 

43.  How  long  is  the  perimeter  of  a  triangle  each  side 
of  which  is  5  inches  ? 

44.  Cut  three  strips  of  paper,  each  7  centimeters  long, 
place  them  so  as  to  inclose  the  largest  possible  triangle, 
and  find  the  length  of  its  perimeter. 

45.  Draw  a  rectangle  and  show  how  many  lines  form 
its  perimeter. 

Note.  —  A  plane  figure  each  of  whose  angles  is  a  right  angle  is 
called  a  Rectangle. 

46.  How  long  is  the  perimeter  of  a  rectangle  of  which 
one  side  is  10  inches  in  length  and  the  side  adjacent  to 
this,  6  inches  ? 

47.  Draw  a  rectangle  whose  width  is  5  centimeters, 
and  whose  length  is  three  times  its  width,  and  find  the 
length  of  its  perimeter. 

48.  Find  the  perimeter  of  a  rectangle  whose  width  is 
6  inches  and  whose  length  is  5  inches  more  than  twice  its 
width. 

49.  Find  the  perimeter  of  a  rectangle  one  side  of  which 
is  12  inches  and  an  adjacent  side  of  which  is  6  inches  less 
than  three  times  as  long.  The  width  of  this  rectangle  is 
what  fractional  part  of  its  length  ? 


14  LINES  AND  ANGLES. 

50.  Draw  a  rectangle  whose  length  is  14  inches  and 
whose  width  is  20  inches  less  than  twice  its  length.  Find 
the  perimeter. 

51.  How  long  is  the  perimeter  of  a  rectangle  whose 
sides  are  each  8  inches  ? 

52.  What  name  is  given  to  a  rectangle  whose  sides  are 
all  equal  ? 

Note.  —  A  plane  figure  having  four  equal  sides  and  four  right 
angles  is  called  a  Square. 

53.  Is  every  square  a  rectangle  ?  Is  every  rectangle 
a  square  ?     Explain  your  answers. 

A 


54.  How  long  is  each   side  of  a  square 
whose  perimeter  is  28  inches  ? 

55.  In  the  square  ABCD,  what  line  has 
the  same  direction  as  AB^f  ^ 

NoTK.  —  Lines  which  have  tlie  same  direction  are  called  Parallel 
Lines. 

50.    What  line  is  parallel  to  ^C7? 

57.  If  the  perimeter  of  the  square  is  24  inches,  how 
far  is  the  line  AB  from  the  line  CD  ?  Are  they  every- 
where equally  distant  ?  If  they  were  both  prolonged, 
would  they  ever  meet  ? 

58.  Draw,  if  you  can,  two  parallel  lines  which  are  3 
inches  apart  in  one  place  and  5  inches  apart  in  another 
place. 

59.  Draw  two  lines  perpendicular  to  each  other.  What 
kind  of  an  angle  is  formed  ? 

60.  In  the  square  which  illustrates  Ex.  55,  what  lines 
are  perpendiciUar  to  CD'i     To  A  (7? 


LINES  AND  ANGLES.  15 

61.  How  long  is  the  perimeter  of  a  square  inch  ? 

62.  How  many  inches  are  there  in  the  perimeter  of 
a  square  foot  ? 

63.  Place  two  pieces  of  paper  or  wood,  each  1  inch 
square,  on  a  flat  surface  so  that  an  edge  of  one  shall 
coincide  with  the  edge  of  the  other,  and  find  the  perimeter 
of  the  rectangle  thus  formed. 

64.  Place  5  one-inch  squares  in  a  row,  their  edges 
coinciding  where  possible,  and  find  the  perimeter  of  the 
rectangle  thus  formed. 

65.  Place  10  inch  squares  in  two  rows  as 
in  the  cut,  and  find  the  perimeter  of  the 
rectangle  thus  formed. 

QQ.  Place  25  one-inch  squares  so  as  to  form  a  square, 
and  find  its  perimeter. 

67.  In  the  square  ABOD.,  which  angles 
are  next  to  the  angle  ^?  Which  angle  is 
not  next  or  consecutive  to  Al  What  kind 
of  a  line  is  the  shortest  distance  from  ^  to  i>  ? 
What  geometric  principle  tells  that  ? 

68.  Reproduce  the  square,  and  draw  a  diagonal  from 
the  point  A. 

Note.  —  A  Diagonal  of  a  plane  figure  is  a  line  drawn  from  the 
vertex  of  one  angle  to  the  vertex  of  another  angle  which  is  not  con- 
secutive. 

69.  How  many  diagonals  can  be  drawn  from  the 
point  A  ?     What  geometric  principle  applies  here  ? 

70.  How  many  diagonals  can  be  drawn  in  a  square  ? 
How  many  can  be  drawn  in  a  rectangle  which  is  not 
a  square  ? 


16  LINES  AND  ANGLES. 

71.  How  many  diagonals  can  be  drawn  in  a  triangle 
Give  the  reason  for  your  answer. 

72.  In  the  triangle  ABC,  which  is  the 
longer,  AC  or  ABC f  What  geometric 
principle  states  that  fact  in  a  general  way  ? 
Which  is  the  longer,  AB  or  AC-{-  CB? 

73.  Show  the  truth  of  the  following  proposition  : 

PRiNcrPLE  8.  — ^4n7/  side  nf  a  trmngle  is  less  than 
the  sum  of  the  other  two  sides. 

74.  Cut  three  narrow  strips  of  paper,  one  10  inches 
long,  the  others  each  5  inches  long,  and  j)lace  them,  if 
you  can,  so  as  to  make  a  triangle. 

76.  Can  you  make  a  triangle,  one  side  of  which  is 
12  inches  and  each  of  the  other  sides  of  which  is  6 
inches  long? 

76.  Can  you  make  a  triangle  whose  sides  are  7  inches, 
8  inches,  and  9  inches  ?  4  inches,'  5  inches,  and  10  inches  ? 
Give  reasons. 


77.    If  AB^l  inches  and   BC=^ 
inches,  how  long  is  AC'^     What  line  a 
is  the  sum  of  the  lines  AB  and  BC'} 


78.  What  line  is  the  difference  between  AC  and  ABl 
Between  AC  and  BC^ 

79.  What   line  equals  the 

sum  oi  AC  and  CD?    Of  BD  j, f ^ b 

and2>e? 

80.  What  line  is  the  difference  between  AB  and  ACf 
Between  AD  and  CD  ?     Between  AB  and  DB  ? 


LINES  AND  ANGLES.  17 

81.  What  line  is  the  difference  between  AB  and  the 
sum  oi  AC  and  CDl 

82.  AD  +  BB-AC='> 

83.  AO+CB-BB=? 

84.  AB-hBB-CB  =  ? 

85.  AO-\-OB-(AB-OB)  =  ? 

86.  ^i)  +  2>.B-(^(74-Ci>)  =  ? 

87.  What  angle  is  the  sum  of  the  adja- 
cent angles  ABO  sind  CBB  ?  What  side 
is  common  to  those  angles  ? 

88.  If  you  take  the  angle  CBB  from  the  angle  ABB^ 
what  angle  remains  ? 

89.  If  one  line  is  5  times  another,  and  the  sum  of  the 
lines  is  18  inches,  how  long  is  each  line  ? 

90.  Lay  off  a  line  3  inches  long  on  a  line  15  inches 
long  and  find  how  many  times  the  longer  line  contains 
the  shorter  one. 

Note.  —  A  line  which  is  contained  in  another  line  a  whole  num- 
ber of  times  is  said  to  measure  it. 

91.  How  many  times  will  a  line  4  inches  long  measure 
a  line  32  inches  long  ? 

92.  How  many  times  will  a  line  1  inch  long  measure 
the  length  of  your  desk  ? 

93.  How  many  times  will  a  stick  6  centimeters  long 
measure  a  stick  48  centimeters  long  ? 

94.  A  coin  which  measures  3  inches  around  the  edge 
will  revolve  how  many  times  in  rolling  12  inches? 

SuGGESTiox.  —  Roll  a  coin  and  find  how  the  distance  around  its 
edge  compares  with  the  distance  it  rolls  in  revolving  once. 

HORX.    GEOM. —  2 


18  LINES  AND  ANGLES. 

95.  How  many  times  will  a  wheel  whose  circumference 
is  6  feet  revolve  in  running  60  feet  ? 

Query.  —  What  is  the  circumference  of  a  wheel  ? 

96.  How  long   is   the   tire   of  a  wheel   that   revolves 

9  times  in  running  45  feet? 

97.  What  is  the  circumference  of  a  coin  that  revolves 

10  times  in  roUing  40  inches  ? 

98.  What  is  the  length  of  the  longest  line  that  can  be 
laid  off  a  whole  number  of  times  on  each  of  two  lines, 
one  6  inches  long  and  the  other  8  inches  long  ? 

Note.  —  The  longest  line  which  exactly  measures  two  or  more 
lines  is  called  their  Greatest  Common  Measure. 

99.  What  line  is  the  greatest  common  measure  of  two 
lines,  one  15  inches,  the  other  18  inches  ? 

100.  How  long  is  the  G.  C.  M.  of  a  12-inch  line  and  an 
18-inch  line  ? 

101.  What  is  the  circumference  of  the  largest  coin 
which,  in  rolling  either  9  inches  or  12  inches,  will  make  a 
numlnir  of  complete  revolutions  ? 

102.  What  is  the  circumference  of  the  largest  wheel 
which  will  make  a  number  of  complete  revolutions  in 
running  20  feet,  and  also  in  running  30  feet? 

103.  What  is  the  circumference  of  the  largest  wheel 
which  will  make  a  number  of  complete  revolutions  in  run- 
ning 25  feet,  and  also  in  running  15  feet? 

104.  What  is  the  length  of  tlie  shortest  line  upon 
which  you  can  lay  off  two  lines  respectively  8  inches 
and  10  inches  a  whole  number  of  times?  How  many 
times  will  it  contain  the  longer  line? 


LINES  AND  ANGLES.  19 

Note.  —  The  shortest  line  which  can  be  exactly  measured  by  two 
or  more  lines  is  called  their  Least  Common  Multiple. 

105.  How  long  is  the  L.  C.  M.  of  an  8-inch  line  and  a 
12-inch  line?  How  many  times  is  it  measured  by  the 
8-inch  line  ? 

106.  Draw  the  line  which  is  the  L.  C.  M.  of  a  line  10 

centimeters  and  8  centimeters. 

107.  Find  the  shortest  line  that  can  be  exactly  meas- 
ured by  each  of  two  lines,  one  of  which  is  9  inches  and 
the  other  12  inches. 

108.  Find  the  L.  C.  M.  of  three  lines,  respectively  8 
inches,  12  inches,  and  16  inches. 

109.  A  front  wheel  of  a  toy  wagon  is  20  inches  in  cir- 
cumference ;  a  hind  wheel  is  30  inches  in  circumference. 
How  far  shall  the  wagon  run  that  both  wheels  shall  have 
made  a  number  of  complete  revolutions  ?     Illustrate. 

110.  A  front  wheel  of  a  toy  wagon  is  3  feet  in  ci^rcum- 
ference  ;  a  hind  wheel  4  feet.  In  traveling  what  distance 
will  both  wheels  make  a  number  of  complete  revolutions, 
and  how  many  will  each  make  ? 

111.  Describe  and  illustrate  everything  which  has  been 
explained  in  the  notes  of  this  chapter. 


CHAPTER  II. 

CIRCLES. 

Note.  —  A  Circle  is  a  plane  figure  wliose  perimeter  is  a  curved 
line,  every  ix)int  of  which  \h  etpially  distant  from  a  point  within, 
called  the  Center.  The  perimeter  of  a  circle  is  called  a  Circum- 
ference. A  Diameter  i^  a  Mtraight  line  pa.s8ing  through  the  center 
and  having  VtotU  ends  in  the  circumference.  A  straight  line  from  the 
center  to  the  circumference  is  called  a  Radius. 

1.  With  a  pair  of  dividers  draw  a  circle  witli  a  radius 
5  inches  long.     How  long  is  its  diameter  ? 

2.  Draw  a  circle  with  a  radius  of  4  inches,  and  draw 
several  diameters.     How  long  is  each  diameter  ? 

3.  With  a  riulius  of  3J  inches  draw  a  circle  on  a  piece 
of  pasteboard.  Cut  it  out  and  measure  the  circumference 
with  a  tape  measure.  You  will  find  that  it  is  about  22 
inches  long.  If  it  were  exactly  22  inches,  the  circumfer- 
ence would  l>e  how  many  times  the  diameter  ? 

Note.  —  When  you  study  demonstrative  geometry,  you  will  find 
another  way  of  proving  the  truth  of  the  following  theorem : 

Principle  4.  —  T?ie  circumfereiice  of  every  circle  is 
3.1410+^  or  V  (nearly)  times  its  diavieter. 

4.  What  is  the  circumference  of  a  circle  whose  diameter 
is  14  inches  ? 

Note.  —  We  use  V  ^  express  the  relation  of  a  circumference  to  its 
diameter. 

5.  What  is  the  circumference  of  a  circle  whose  radius 
is  14  inches  ? 

20 


CIRCLES.  21 

6.  What  is  the  circumference  of  a  circle  whose  diam- 
eter is  10  inches  ? 

7.  What  is  the  circumference  of  a  circle  whose  diam- 
eter is  28  meters  ? 

Query.  —  How  many  inches  long  is  a  meter? 

8.  The  radius  of  a  circle  is  2  meters.     Find  its  cir- 
cumference in  centimeters. 

9.  The  diameter  of  a  circle  is  3  meters.     Find  its 
circumference  in  centimeters. 

10.  The  circumference  of  a  circle  is  44  inches.  Find 
its  diameter. 

11.  The  circumference  of  a  circle  is  33  inches.  What 
is  its  radius  ? 

12.  The  circumference  of  a  circle  is  22  inches.  Find 
its  diameter. 

13.  The  diameter  of  a  nickel  of  the  issue  of  1866  is  2 
centimeters.     What  is  the  circumference  ? 

14.  A  horse  is  tied  to  a  stake  by  a  rope  7  feet  long. 
Find  the  length  of  the  longest  path  he  can  travel  in  going 
once  around  the  stake. 

15.  If  the  outer  edge  of  a  merry-go-round  is  10^  feet 
from  the  center,  how  far  does  John  travel  when  he  sits  at 
the  outer  edge  and  makes  one  revolution  ? 

16.  If  he  faces  forwards  and  keeps  his  hands  at  his 
sides,  which  hand  travels  the  faster,  his  right  or  left 
hand  ? 

17.  What  is  the  length  of  the  longest  stick  that 
can  lie  wholly  on  a  round  table  which  is  11  feet  in 
circumference  ? 


22  CIRCLES. 

18.  What  is  the  longest  line  that  can  be  drawn  across 
a  round  flower  bed  whose  circumference  is  12  feet  10 
inches  ? 

19.  Mary  has  embroidered  a  round  mat  1  foot  10  inches 
in  circumference.  If  it  were  cut  into  two  equal  parts, 
what  would  be  the  length  of  the  straight  edge  of  each  ? 

20.  Using  "semi"  to  denote  **half,"  how  would  you 
define  a  semicircle?     A  semicircumference  ? 

21.  Draw  a  semicircle,  and  write  the  names  of  its 
bounding  lines. 

22.  The  radius  of  a  certain  circle  is  2  feet  11  inches. 
By  how  much  does  the  curved  lx)undary  line  of  one  of  its 
semicircles  exceed  the  straight  boundary  line  ? 

28.  The  smaller  circle  is  placed  upon 
the  larger,  so  that  their  centers  coincide. 
If  the  diameter  of  the  larger  circle  were 
10  inches,  and  that  of  the  smaller  7  inches, 
what  would  be  the  width  of  the  circular 
ring  included  between  the  circumferences  ? 

24.  If  yon  cut  a  circle  whose  diameter  is  7  inches  from 
the  center  of  a  circle  whose  diameter  is  11  inches,  what  is 
the  width  of  the  circular  ring  that  remains  ? 

25.  Find  the  length  of  the  arc  which  is  J  of  the 
circumference  of  a  certain  circle  whose  diameter  is  42 
inches. 

Note.  —  An  Arc  is  a  part  of  a  circumference.  Compare  arc  and 
arch.     Is  a  semicircumference  an  arc? 

26.  One  arc  of  a  circle  is  5  times  the  remaining  arc. 
The  circumference  is  120  meters.  How  many  meters  long 
is  each  arc  ? 


CIRCLES.  23 

27.  Find  the  length  of  an  arc  which  is  f  of  the  circum- 
ference of  a  circle  whose  radius  is  14  inches. 

28.  An  arc  is  9  times  the  remaining  arc.  The  diameter 
of  the  circle  is  7  feet.     Find  the  length  of  each  arc. 

Qu^RY.  ■ —  How  long  is  the  cii-curaf erence  ? 

29.  If  a  man  is  on  a  plain  where  he  can  see  3  miles  in 
every  direction,  what  is  the  length  of  the  line  which 
bounds  the  part  of  the  plain  over  which  he  can  see  ? 

30.  Draw  an  arc  and  connect  its  ends  by  a  straight  line. 
The  straight  line  is  called  a  chord. 

Note.  —  The  word  "  subtend "  is  derived  from  two  Latin  words, 
meaning  "to  stretch  under."  Would  you  say  that  the  arc  subtends 
the  chord,  or  the  chord  subtends  the  arc  ? 

31.  Draw  a  segment,  and  write  upon  it  the  names  of  its 
bounding  lines. 

Note.  —  The  part  of  a  circle  included  between  a  chord  and  its  arc 
is  caUed  a  Segment. 

32.  Is  a  semicircle  a  segment  ?     Give  reasons. 

33.  How  many  segments  are  formed  in  a  circle  by 
drawing  a  chord  ?     How  many  arcs  ? 

Note.  —  A  segment  greater  than  a  semicircle  is  called  a  greater 
segment.     A  segment  less  than  a  semicircle  is  called  a  less  segment. 

34.  Can  three  greater  segments  be  cut  from  a  circle  ? 

35.  Which  is  the  longer  line,  an  arc  or  its  chord  ? 
State  the  geometric  principle  which  applies. 

36.  In  the  circle  whose  center  is  0, 
which  is  the  longer,  the  straight  line 
AB  or  the  sum  of  the  lines  AO  and 
OQl  The  straight  line  AC  or  the  sum 
of  the  lines  AO  and  001  The  line 
AO  OT  AB  ?     Give  reasons. 


24  CIRCLES, 

37.  Can  j^ou  draw  a  chord  which  is  not  a  diameter  of 
the  circle  and  yet  is  equal  to  a  diameter  ? 

38.  Can  you  see  the  truth  of  the  following  statement  ? 

pRiNcrPLB  5.  — A  diameter  is  longer  than  any  other 
chord. 

39.  What  is  the  longest  line  that  can  be  drawn  on  the 
head  of  a  drum  40  inches  in  circumference  ? 

40.  An  object  which  has  a  circle  for  its  base, 
and  tapers  to  a  point,  is  called  a  Cone.  Draw 
or  make  a  cone.  What  is  the  length  of  the 
longest  line  that  can  be  drawn  across  its  base  if 
the  circumference  is  121  millimeters? 

41.  Draw  in  a  circle  several  parallel 
chords  on  the  same  side  of  the  center  of 
the  circle.  Cut  out  and  fold  the  circle  so 
as  to  place  one  chord  upon  the  other. 
Which  is  the  longest  ? 

42.  Draw  in  a  circle  several  cliords  from  the  same 
point.     How  do  they  compare  in  length  ? 

48.    Can  you  show  the  truth  of  the  following  statement  ? 

Principle  6.  —  In  the  same  circle  or  in  eguaZ  cir- 
cles  tJie  greater  of  two  chords  is  at  a  less  distance  front 
the  center, 

44.  With  a  given  point  as  a  center  and  with  any  radius 
describe  a  circle  and  cut  it  out.  Draw  a  diameter,  fold 
tlie  circle  along  that  line,  and  see  if  the  two  surfaces 
coincide.     Does  the  diameter  bisect  the  circle  ? 

Note.  —  To  bisect  means  to  divide  into  two  equal  parts. 

45.  Draw  two  parallel  cliords  in  a  circle,  fold  the  circle 
so  that  the  two  halves  of  one  chord  shall  coincide.     Will 


CIRCLES.  25 

the  halves  of   the  other  chord  coincide  ?     Will  the  two 
arcs  intercepted  by  the  ends  of  the  chords  also  coincide  ? 

46.  Show  by  another  circle  the  truth  of  the  following 
statement : 

Principle  7.  — Parallel  chords  intercept  equal  arcs. 

47.  Draw  a  circle  and  a  straight  line  touching  the  cir- 
cumference at  one  point. 

Note.  —  A  straight  line,  which,  however  prolonged,  touches  a 
circumference  at  only  one  point  is  called  a  Tangent,  and  the  point  at 
which  it  touches  is  called  the  point  of  tangency. 

48.  Draw  two  parallel  tangents 
and  join  their  points  of  tangency. 
Try  to  draw  three  parallel  tangents 
to  the  same  circle. 

49.  Draw  three  tangents  to  a  cir- 
cle, placing  two  of  them  upon  one 
semicircumference  and  the  third  on  the  other.     Prolong 
them  until  they  inclose  a  surface.     What  figure  is  formed  ? 

Note.  —  A  plane  figure  whose  sides  are  all  tangent  to  a  circle  is 
said  to  be  circumscribed  about  the  circle  and  the  circle  is  said  to  be 
inscribed  in  the  figure. 

50.  Try  to  draw  three  tangents  to  a  circle  in  such  a 
way  that  when  prolonged  they  form  a  triangle  which  is 
not  circumscribed  about  the  circle. 

51.  Draw  four  tangents  to  a  circle,  placing  two  of 
them  on  each  semicircumference,  and  prolong  them  until 
they  inclose  a  surface.     What  figure  is  formed  ? 

XoTE.  —  A  plane  figure  bounded  by  four  straight  lines  is  called  a 
Quadrilateral. 

52.  Draw  two  parallel  tangents  to  a  circle  and  another 
pair  of  parallel  tangents  perpendicular  to  the  first,  and 


26  CIRCLES. 

write  the  name  of  the  figure  thus  circumscribed  about  the 
circle. 

63.  Can  you  circumscribe  about  a  circle  a  rectangle 
which  is  not  square  ? 

64.  Describe  everything  which  is  explained  in  the  notes 
of  this  chapter,  forming  a  clear  mental  picture  of  that 
which  you  describe. 


CHAPTER  III. 
ARCS  AND  ANGLES. 

Note  to  Teacher.  —  A  fan  opened  so  as  to  form  different  angles 
is  very  useful  in  illustrating  the  subject  of  this  chapter,  parts  of  the 
edge  of  the  fan  being  used  to  represent  arcs. 

The  circumference  of  every  circle  may  be  considered 
as  divided  into  360  equal  parts  called  degrees,  marked  °. 

1.  Draw  a  semicircumference  and  tell  how  many 
degrees  there  are  in  it. 

2.  How  many  degrees  in  the  circumference  formed  by 
the  edge  of  a  teacup  ?  How  many  in  the  Equator  ? 
Which  is  the  longer,  a  degree  of  the  circumference  of 
the  teacup  or  a  degree  of  the  Equator  ? 

3.  An  arc  of  one  degree  is  what  fractional  part  of  the 
circumference  ? 

4.  How  long  is  an  arc  of  one  degree  of  a  circum- 
ference which  is  1800  inches  ? 

5.  How  many  degrees  are  there  in  the  arc  described 
by  the  end  of  the  minute  hand  of  a  clock  in  half  an  hour  ? 

6.  How  many  degrees  are  there  in  a  quadrant  ? 
Note.  —  A  Quadrant  is  one  fourth  of  a  circumference. 

7.  In  3  hours,  how  many  degrees  are  described  by  the 
hour  hand  of  a  clock  ?  How  many  by  the  minute  hand  in 
10  minutes  ? 

27 


r^, 


28  ARCS   AND  ANGLES. 

8.  How  many  degrees  are  described  by  the  second 
hand  of  a  watch  in  |  of  a  minute? 

9.  How  many  degrees  are  described  by  the  minute 
hand  in  20  minutes? 

10.  Jolin  and  James  started  from  the  same  point  in  a 
circular  race  track,  and  ran  around  the  track  in  opposite 
directions  until  they  met.  John  ran  -f^  of  the  distance. 
How  many  degrees  were  there  in  the  arc  which  each 
passed  over? 

11.  The  arc  ANB  is  twice  the  arc  AMB, 
How  many  degrees  are  there  in  each  ? 

12.  Ha  circumference  is  divided  by  a  i  , 
chord  so  tliat  the  greater  arc  is  3  times  the  V  / 
less,  how  many  degrees  are  there  in  each  arc  ?       ^ — 

18.  U  the  angles  ABD  and  DBO 
are  equal,  the  lines  BD  and  A  C  are  in 
what  relative  iMJsition?  What  kind  of 
angles  are  ABD  and  BBC?  ^— 

SuGOKSTio.N.  — See  Chap.  1,  Ex.  31,  Note. 

14.  At  what  times  of  day,  when  the  minute  hand  is  at 
XII.,  do  the  hands  of  a  clock  form  a  right  angle  ? 

15.  How  many  degrees  are  there  in  the  angle  formed 
by  the  hands  of  a  clock  at  3  o^clock  ? 

Note.  —  A  right  angle  is  an  angle  of  90  degrees. 

16.  How  many  degrees  are  there  in  the  angle  formed 
by  the  hands  of  a  watch  at  1  o'clock  ? 

17.  How  many  degrees  are  there  in  the  angle  formed 
by  the  hands  of  a  clock  at  4  o'clock  in  the  morning  ?  At 
4  in  the  afternoon  ? 


ARCS  AND  ANGLES.  29 

18.  How  many  degrees  are  there  in  the  angle  formed 
by  the  hands  of  a  clock  at  10  o'clock  ? 

19.  How  many  degrees  are  there  in  the  angle  formed 
by  the  hands  of  a  watch  at  8  o'clock? 

20.  At  what  time  of  the  day,  when  the  minute  hand  is 
at  12,  do  the  hands  form  a  straight  angle  ? 

Note.  —  An  angle  equal  to  two  right  angles  is  called  a  Straight 
Angle.  Each  of  the  lines  which  form  it  is  the  prolongation  of  the 
other. 

21.  Draw  the  line  CD  perpendicu- 
lar to  AB,  as  in  the  cut.  How  many 
degrees  are  there  in  the  two  angles 
ABC  eiiid  CBB?  ^ 


-^ 


22.  If  the  line  01)  were  inclined  as 
in  the  cut,  how  many  degrees  would 
there   be   in   the   sum    of   ADQ  and 
CDB'>     If  ABC  is  120°,  how  many    ^ 
degrees  is  CBB  ? 

Query.  —  As  one  angle  grows  larger  and  the  other  smaller,  does 
the  number  of  degrees  in  their  sum  change  ? 

Principle  8.  —  The  sum  of  the  two  adjacent  angles 
formed  hy  one  straight  line  meeting  another  is  equal 
to  two  right  angles. 

23.  What  is  the  supplement  of  an  angle  of  100°  ? 

Note.  —  Two  angles  which  together  equal  180°  are  said  to  be 
Supplements  of  each  other.  Two  arcs  which  together  equal  a  semi- 
circumference  are  Supplements  of  each  other. 

24.  What  is  the  supplement  of  a  right  angle  ? 

25.  What  is  the  supplement  of  an  angle  which  is  |^  of  a 
right  angle  ? 


80  ARCS  AND  ANGLES. 

26.  What  is  the  supplement  of  an  arc  of  75®  ? 

27.  What  is  the  supplement  of  an  angle  of  47°  ? 

28.  What  is  the  supplement  of  an  arc  of  68°  ? 

29.  If  the  angle  ABC  is  50°,  how  many 
degrees  are  in  the  angle  formed  by  prolong- 
ing the  line  AB  from  the  point  J5?  By 
prolonging  CB  from  the  point  B?  a- 

30.  Draw  by  means  of  your  protractor  an  angle  of  70°, 
and  show  how  many  degrees  its  supplement  has. 

81.  Measure  the  angles  formed  by  the  branching  from 
the  main  stem  of  different  plants,  and  determine  the  sup- 
plements of  those  angles. 

32.  How  many  degrees  has  an  angle  whose  supplement 
is  3  times  as  large  as  the  given  angle  ? 

33.  How  many  degrees  has  an  angle  whose  supplement 
is  95°?  75°?  1°?  5J°? 

34.  How  many  degrees  are  there  in  an  angle  which  is  4 
times  its  supplement  ? 

35.  Can  a  right  angle  have  an  obtuse  or  an  acute  angle 
for  its  supplement  ?     Give  reasons. 

86.  Is  the  supplement  of  an  obtuse  angle  obtuse  or 
acute? 

37.    How  many  degrees  are  there  in 
the  sum  of  the  angles  ABC  and  CBD  ? 
Is  the  number  of  degrees  in  the  angle  a- 
AB  C  changed  by  drawing  the  line  EB  ? 
How  many  degrees  are  there  in  the  sum  of  the  angles 
ABKUBCimd  CBD? 


ARCS  AND  ANGLES. 


31 


38.  a-\-b-^c+d  +  e  =  how   many 
degrees?    Give  reasons. 

Principle  9.  —  The  sum  of  all  the  angles  formed  at 
a  given  point  on  the  same  side  of  a  straight  line  is 
equal  to  two  right  angles  or  180°. 

39.  Angle  BBF  is  a  right  angle. 
Angle  BBE  =  EBF. 
Angle  ABO  ==2  times  CBD.     '  ^■ 
How  many  degrees  are  there  in  each  angle  ? 

40.  Place  the  vertex  of  a  right  angle  at  a  point  in  the 
line  AB  so  that  neither  of  its  sides  shall  coincide  with 
that  line.  How  many  degrees  are  in  the  sum  of  the  two 
angles  formed  by  AB  and  the  sides  of  the  right  angle  ? 

41.  Angle  ACrD  =  a  right  angle. 
Angle  BaU=2  times  angle  UGF. 
Angle  CaB  =  angle  B  GE.  b^  ' 
AnglQAGB  =  angle  BGC. 
How  many  degrees  are  there  in  each  angle  ? 

42.  How  many  degrees  has  each  of  three  angles  formed 
at  a  given  point  on  the  same  side  of  a  straight  line  if  the 
first  contains  5  times  as  many  degrees  as  the  second, 
and  the  third  contains  4  times  as  many  as  the  second? 
Illustrate. 

43.  How  many  degrees  are  there  in  all  the  angles 
formed  on  both  sides  of  the  line  AB  at  the  point  (7?  At 
any  other  point  in  the  line  AB  ? 

44.  How  many  degrees  are  there  in  all  the  angles 
formed  around  a  common  point  ? 


82 


ARCS  AND  ANCLES. 


4.5.  AB  and  CI)  are  diameters  cross- 
ing at  right  angles.  How  many  degrees 
are  there  in  each  of  the  four  arcs  inter- 
cepted by  them  ? 

SuGOKSTiox.  —  Cut  out  a  similar  tigure  and, 
folding  it,  place  the  parts  one  upou  another 
and  see  if  they  coincide. 

46.  If  the  angle  AOC  is  bisected  by  a  radius,  how 
many  degrees  has  each  of  the  angles  thus  formed  ?  How 
many  has  each  of  the  arcs  thus  formed?  Illustrate, 
using  a  protractor,  or  folding. 

Note.  —  It  will  he  seen  that  the  number  of  degrees  in  the  arc  is 
the  same  as  the  num1>er  of  degrees  in  the  angle  at  the  center,  whose 
sides  touch  the  ends  of  the  arc.  or,  as  is  said, 
*'  the  angle  is  measured  by  the  intercepted  arc." 

47.  How  many  degrees  are  there  in 
arc^C?  In  arc  CD?  In  arc  i>jP?  In 
arc  FB? 

Pkinciple  10.  — ^n.  angle  at  the 
center  of  a  circle  U  measured  by  the 
arc  ifUercepted  by  tJie  radii  which  form  it. 

Note.  —  An  angle  at  the  center  is  called  a  Central  Angle. 

48.  Draw  a  circle  and  make  an  angle  of  120°  by  radii. 
The  arc  intercepted  by  the  radii  is  what  part  of  the 
circumference  ? 

40.  An  arc  intercepted  by  two  radii  is  J  of  the  remain- 
ing arc  of  the  circumference.  What  angle  is  formed  by 
the  radii  ? 

50.  A  certain  circumference  consists  of  two  arcs,  one  of 
which  is  8  times  the  other.  How  many  degrees  are  there 
in  the  angle  measured  by  the  smaller  arc  ?     Illustrate. 


ARCS  AND  ANGLES.  33 

51.  A  quadrant  is  divided  into  two  arcs,  one  of  which 
is  3  times  the  other.  How  many  degrees  are  there  in 
each  angle  measured  by  them  ? 

Note.  —  Two  arcs  whose  sum  is  a  quadrant  are  said  to  be  Com- 
plements  of  each  other.  Two  angles  whose  sum  is  a  right  angle  are 
said  to  be  Complements  of  each  other. 

52.  What  is  the  complement  of  an  angle  of  75°  ? 
Of  8J°? 

53.  Draw  the  complement  of  an  angle  of  40°,  and  the 
complement  of  an  arc  of  50°. 

54.  Which  is  the  greater,  an  angle  or  its  complement  ? 
Explain.  Which  is  the  greater,  an  angle  or  its  supple- 
ment ? 

55.  By  how  much  does  the  supplement  of  an  angle  of 
70°  exceed  its  complement  ?  An  angle  of  80°  ?  Any 
angle  of  less  than  90°  ? 

56.  If  AB  and  OJD  are  perpendicular  to  each  other, 
what  part  of  the  circle  is  the  plane  fig- 
ure 0  OB  ?  Cut  out  and  fold  a  circle  and 
shoAV  the  truth  of  your  answer.  What 
are  the  boundaries  of  OCBl  The  arc 
CB  is  what  part  of  the  circumference  ? 

Note.  —  A  surface   inclosed  by  two  radii 
and  their  intercepted  arc  is  called  a  Sector. 

57.  Draw  a  circle,  divide  it  into  six  equal  sectors  by 
making  angles  of  60°  at  the  center,  and  show  what  part 
the  arc  of  each  sector  is  of  the  circumference. 

58.  The  arc  of  a  sector  which  is  -I  of  a  circle  is  what  part 
of  the  circumference  ?     How  many  degrees  has  its  angle  ? 

59.  What  have  you  ever  eaten  whose  upper  surface 
could  be  represented  by  a  sector  ? 

HORN.   GEOM. 3 


34  ^RCS  AND  ANGLES. 

60.  Make  a  sector  whose  angle  is  40°  and  whose  straight 
edges  are  7  inches.  What  part  of  a  circle  is  it?  Find 
the  length  of  the  circumference  of  the  circle.  The  length 
of  the  arc.     The  length  of  the  perimeter  of  the  sector. 

61.  A  semicircle  is  divided  into  two  sectors,  the  arc 
of  one  being  4  times  that  of  the  other.  How  many 
degrees  are  there  in  the  arc  of  each  sector  ?  The  smaller 
sector  is  what  part  of  the  semicircle  ? 

62.  A  sector  whose  arc  is  150**  is  divided  into  two 
sectors,  the  arc  of  one  being  3  times  that  of  the  other. 
How  many  degrees  are  there  in  the  angle  of  each  ?  Each 
sector  is  what  fractional  part  of  the  circle  ? 

63.  How  long  is  the  perimeter  of  a  sector  of  120*  in  a 
circle  whose  radius  is  10 J  feet  ? 

64.  Find  the  diflference  between  the  length  of  the 
curved  boundary  line  and  the  sum  of  the  two  straight 
boundary  lines  of  a  sector  of  160**  in  a  circle  whose  diame- 
ter is  17^  inches. 

65.  If  a  round  pie  is  cut  into  six  equal  pieces,  how 
many  degrees  has  the  arc  formed  by  the  curved  edge  of 
the  crust  of  each  piece  ?  How  many  degrees  has  the 
angle  formed  by  the  straight  sides  of  each  piece  ? 

66.  How  many  times  will  an  arc  of  20"  be  contained  in 
an  arc  of  80**  of  the  same  circumference  ? 

67.  How  many  times  will  an  arc  whose  length  is  20 
inches  be  contained  in  a  circumference  whose  length  is 
5  feet. 

68.  Make  a  sector  of  90**  and  show  by  folding  how  many 
times  it  will  contain  a  sector  of  45**  of  the  same  circle. 

69.  A  wheel  which  has  6  cogs  is  geared  with  one  that 


ARCS  AND  ANGLES.  35 

has  60  cogs.     How  many  times  will  the  small  wheel  turn 
while  the  large  one  turns  once  ? 
Suggestion.  —  Examine  clockwork. 

70.  What  name  is  given  to  a  quantity  which  measures 
another  quantity  a  whole  number  of  times  ?  To  the  quan- 
tity which  is  an  exact  measure  of  two  or  more  quantities  ? 
To  the  largest  quantity  which  exactly  measures  two  or 
more  quantities  ? 

71.  How  many  degrees  are  there  in  the  arc  which  is 
the  G.  C.  M.  of  an  arc  of  70°  and  one  of  90°  degrees  in 
the  same  circle  or  equal  circles  ? 

72.  One  arc  of  a  circle  is  21  inches  ;  another  arc  of  the 
same  circle  is,  28  inches.  Find  the  length  of  the  arc 
which  is  their  greatest  common  measure. 

73.  What  sector  is  the  G.  C.  M.  of  two  sectors  of  the 
same  circle,  one  of  60°,  the  other  of  150°  ? 

74.  Draw  an  angle  of  40°  and  another  of  50°  with 
lines  5  inches  long,  and  show  how  many  degrees  there  are 
in  their  G.  C.  M. 

75.  What  name  is  given  to  a  quantity  which  contains 
another  quantity  a  whole  number  of  times  ?  To  the 
quantity  which  is  an  exact  multiple  of  two  or  more  quan- 
tities ?  To  the  least  quantity  which  is  an  exact  multiple 
of  two  or  more  quantities  ? 

76.  What  arc  is  the  L.  C.  M.  of  an  arc  of  10°,  and  of 
an  arc  of  15°  of  the  same  circle  ? 

77.  How  many  degrees  has  the  arc  of  the  sector  which 
is  the  L.  C.  M.  of  two  sectors  of  the  same  circle,  the  arc 
of  one  being  8°  and  of  the  other  6°  ? 


86  ARCS  AND  AXGLES, 

78.  What  is  the  L.  C.  M.  of  an  angle  uf  15°,  and  of  an 
angle  of  25°  ? 

79.  Find  the  L.  C.  M.  of  an  arc  of  90**  and  one  of  120° 
of  the  same  circle. 

80.  A  front  wheel  of  a  wagon  is  21  inches  in  diameter, 
a  hind  wheel  28  inches  in  diameter.  How  far  can  the 
wagon  run  that  both  wheels  may  make  a  number  of  com- 
plete revolutions,  and  how  many  will  each  make  ? 

81.  Show  that  a  semicircle  is  a  sector  and  a  segment. 

82.  If  the  sector  ABC  is  superposed  U|x>n  the  equal 
sector  DEF  so  that  the  |>oint 
A  coincides  with  the  point 
2),  B  with  F,  and  C  with  E, 
would  the  chord  AB  coin- 
cide with  the  chord  DF'f 
Can  you  draw  a  straight  line 
between  the  [xiints  2>  and  F  that  would  not  coincide 
throughout  its  whole  extent  with  the  line  DF'i  Quote 
a  geometric  principle  which  is  applicable. 

88.  Divide  a  circumference  into  four  equal  arcs  by 
diameters.  Join  the  extremities  of  each  arc  by  a  chord. 
Superpose  and  show  the  truth  of  the  following  principle; 

Principle  11. — In  tlie  same  circle  or  in  equal  cir- 
cles equal  arcs  are  subtended  by  equal  chords. 

^  84.    Given  the  angle  ABC  W  ;  how 
many  degrees  has  the  angle  fonned  by 

prolonging    CB  to   2>?     By   prolong-   q 2^. ^ 

ing  AB  to  J^?     Which  angle  is  the  X, 

greater,  ABD  or   CBE'^     How  many  \| 

degrees  are  there  in  the  angle  DBE  ?     Why  ? 


ARCS  AND  ANGLES. 


37 


85.  Looking  along  the  line  ABE^  what  angle  is  the 
supplement  of  ABD  ?  Looking  along  CBD^  what  is  the 
supplement  of  ABD  ?     Are  those  angles  then  equal  ? 

86.  If  ABD  =  lbQ%  what  is  BBE"^     What  is  ABC^ 

87.  If  the  angle  CBE  equaled  130°,  how  many  degrees 
would  there  be  in  each  of  the  other  angles  ? 

88.  Draw  two  intersecting  lines  and  show  which  angles 
are  equal. 

Pkinciple  12. — If  two  straight  lines  intersect,  the 
vertical  or  opposite  angles  are  equal. 


89.    Angle  a  =  twice  angle  b. 
angles  a,  5,  c,  d. 


Find 


90.    Angle  a  =  50°.    Angle  c  =  50°. 
Find  each  of  the  others. 


91.  Angle  x  =  90°.  Find  each 
angle  around  the  common  vertex. 
Angle  a  =  3  times  angle  m. 

92.  Draw  a  line  AB^  and  at  some 
point,  as  6^,  make  an  angle  of  70°.  At 
some  other  point,  as  i),  make  another 
angle  of  70°.  Since  CE  and  BF  di- 
verge from  AB  at  the  same  angle,  will 
they  lie  in  the  same  direction  ?  Are 
they  parallel  ?  How  many  degrees  are 
there  in  tjie  angle  EOB  ?     In  FBA  ? 


38 


ARCS  AND  ANGLES, 


93.  Draw  two  parallels  l)ranching 
from  another  line  AB,  making  the 
angle  A  CD  80°.  How  many  degrees 
has  the  angle  CUF? 

Suggestions. —  1.  Cut  out  angle  A  CD  and 
superpose  it  upon  angle  CEF. 

2.  Measure  the  angles  with  a  protractor. 

94.  How  many  angles  are  formed 
by  a  transversal  to  two  parallels? 

Note.  —  A  line  crossing  or  meeting  two 
parallels  is  called  their  Transversal.  The 
angles  outride  the  parallels  are  called  Ex- 
terior Angles ;  those  within  the  parallels  are 
called  Interior  Angles. 

95.  Name  the  interior  angles  on  the  right  side  of  the 
transversal.     On  the  left  side. 

96.  Name  the  exterior  angles  on  the  right  side  of  the 
transversal.     On  the  left  side. 

Note.  —  Angles  on  different  sides  of  the  transversal  which  are 
not  adjacent  are  ealled  Alternate  Angles. 

97.  Name  the  two  pairs  of  alternate  interior  angles. 

98.  Name  the  two  pairs  of  alternate  exterior  angles. 

Note.  —  Two  angles  which  are  on  the  same  side  of  the  trans> 
versal,  one  interior  and  the  other  exterior,  but  not  adjacent,  are  called 
Opposite  Interior  and  Exterior  Angles. 

99.  Angle  EGB  being  an  exterior,  what  is  its  opposite 
interior  angle  ?  Name  the  exterior  angle  opposite  to  angle 
A  Gff,  Name  the  exterior  angle  opposite  to  angle  HGB, 
Name  the  interior  angle  opposite  to  angle  A  Q-E. 

100.  Can  you  draw  two  parallels  cut  by  a  transversal 


ARCS  AND  ANGLES.  39 

in  such  a  way  that  an  exterior  angle  shall  be  80°  and  its 
opposite  interior  angle  shall  be  70°  ? 

101.  Reproduce  the  figure  in  Ex.  94,  changing  it  so 
as  to  make  angle  EG-B  60°.  How  many  degrees  are  there 
in  angle  (^^i>? 

Principle  13.  —  When  two  parallel  lines  are  cut  hy 
a  transversal,  the  opposite  interior  and  exterior  angles 
are  equal. 

102.  If  angle  JEGrB  is  50°,  how  many  degrees  are  there 
in  the  angle  BGHl  Quote  the  geometric  principle  by 
which  you  can  tell  the  number  of  degrees  in  angle  BG-H 
when  the  number  in  angle  BGB  is  given. 

103.  Angle  UGB  being  50°,  how  many  degrees  has  the 
angle  AGIT?     Quote  the  geometric  principle. 

104.  Angle  UGB  being  50°,  how  many  degrees  has  angle 
AG-U?     Angle  GHB?     Quote  the  geometric  principles. 

105.  If  angle  A  GE  were  100°,  how  many  degrees  would 
there  be  in  each  of  the  eight  angles  formed  by  the  two 
parallels  and  their  transversal  ? 

106.  If  angle  AGE  is  3  times  the  angle  EGB,  how 
many  degrees  are  there  in  each  angle  formed  by  the  three 
lines  ? 

107.  Draw  a  horizontal  line  and  mark  two  points  in  it. 
At  these  points  draw  lines  crossing  the  horizontal  in  the 
same  direction,  making  an  angle  of  75°  with  it.  How 
many  degrees  has  each  angle  formed  by  the  three  lines  ? 
How  does  the  figure  differ  from  that  given  in  Ex.  94  ? 

108.  Draw  two  parallels  and  a  transversal  so  that  an 
exterior  angle  shall  be  4  times  its  adjacent  interior  angle. 
How  many  degrees  has  each  of  the  eight  angles  ? 


40  ARCS   AXD  ANGLES. 

109.  John  and  William  with  their  sister  Mary  went 
on  an  excursion.  Their  father  provided  each  of  his  chil- 
dren with  money  so  that  John's  money  equaled  Mary's 
money  and  William's  money  equaled  Mary's  money. 
Which  of  the  children  had  the  more  money.  John  or 
William  ? 

110.  If  the  following  axiom  seems  true  to  you,  illus- 
trate it  by  three  angles. 

Axiom  1. 

Things  which  are  equal  to  the  %ame  thing  are  equal  to 
each  other. 

111.  AB  and  CD  being  parallel,  how  many  degrees 
are  there  in  angle  y?  Give  the  rea- 
son. How  many  in  angle  x*(  Give 
the  reason.  Which  is  the  greater 
of  the  two  alternate  interior  angles, 
X  {}V  yl  Which  is  the  greater  of  the 
two  alternate  exterior  angles,  the 
angle  which  is  70®  or  angle  m  ? 

PuiNcrPLE  14.  —  When  two  parallels  are  cut  by  a 
trarisversal,  the  alterfiale  interior  angles  are  equal. 

112.  AB  and  CD  are  parallel.    The 

angle  /  being  twice  the  angle  c,  how  \^ 

many  degrees   has   the   angle   c'i     If     a — ^!^^t b 

angle  a  were  a  right  angle,  how  many     o ^^ — ^ 

degrees  would  there  be  in  each  of  the  \ 
angles  formed  by  the  three  lines  ? 

118.  If  the  transversal  is  perpendicular  to  one  of 
two  parallels,  what  angles  will  it  form  with  the  other  ? 
Illustrate. 


ARCS  AND   ANGLES. 


41 


B     C 


114.  If  ^  =  70°,  how  many  degrees 
are  there  in  /  ?  How  many  in  g  ?  If 
a  =  110°,  how  many  degrees  are  there 
in  h  ?     How  do  b  and  g  compare  ? 


115.  If  angle  d  is  135°,  how  many 
degrees  are  there  in  each  of  the  re- 
maining seven  angles  ? 

116.  Let  the  parallels  AU  and  BF 
meet  the  line  01).  If  the  angle  BFB 
=  112°,  how  many  degrees  has  each 
of  the  remaining  angles  formed  by  the 
three  lines  ? 


A     D 


117.  AB  and  CI)  are  parallel. 
Angle  b  =  63°.  How  many  degrees 
are  there  in  each  of  the  other  seven 
angles  formed  by  the  three  lines  ? 

118.  Let  the  parallels  AB  and  01) 
be  crossed  by  the  parallels  BGr  and 
FH.  Angle  a  =  100°.  How  many 
degrees  has  each  of  the  other  15 
angles?  How  many  are  there  in  the 
sum  of  angle  /  and  angle  k  ?  How 
many  degrees  are  there  in  the  sum  of 
angle  g  and  angle  I  ?  In  the  sum  of 
angle  h  and  angle  m  ? 
In  the  sum  of  angle  e 
b  and  angle  c  ? 


In  the  sum  of  angle  o  and  angle  p  ? 
and  angle  j?     In  the  sum  of  angle 


42 


ARCS  AND  ANGLES, 


119.  AB   and    CD   are    horizontal 
lines. 

U  b  =  75°,  how  many  degrees  are    ^ 
there  in  the  sum  of  d  and  /?  c 

If   i  =  86°,  how  many  degrees  are 
there  in  the  sum  of  d  and  /? 

If  h  =  85°,  how  many  degrees  are  there  in  the  sum  of 
d  and  /? 

If  i  =  77 J°,  how  many  degrees  are  there  in  the  sum  of 
d  and/? 

120.  When  two  parallel  lines  are  crossed  by  a  trans- 
versal, what  is  the  sum  of  the  two  interior  angles  on  the 
same  side  of  the  transversal  ? 

Principle  15.  —  WTien  two  parallels  are  cut  by  a 
transversal,  the  sum  of  the  interior  angles  on  tli^  same 
side  is  equal  to  tivo  right  angles. 

121.  When  the  angle  /  is  twice  the 
angle  g^  how  many  degrees  has  each  of 
the  interior  angles  ? 

122.  When  angle/  is  150°,  how  many 
degrees  has  each  of  the  angles  formed  by 
the  three  lines  ? 


123.    AB   and    CD   are   parallel.       Angle 
A8(7=107°.     Find  angle  BCD. 


124.    AB    and    CD    are    parallel.     Angle 
BCD  =  86°.     Find  angle  ABC. 


CHAPTER   IV. 
CUMULATIVE  REVIEW  NO.  1. 

1.  Define  each  term  in  the  following  classification  : 

\  Right. 

^"^^^^     Oblique  I  ^^^'^• 
*•  ^      \  Obtuse. 

2.  Draw  two  adjacent  angles,  two  adjacent  supple- 
mentary angles,  two  supplementary  angles  that  are  not 
adjacent,  two  adjacent  complementary  angles,  two  com- 
plementary angles  that  are  not  adjacent. 

3.  Can  you  draw  two  unequal  vertical  angles  ? 

4.  Define  perimeter.  What  special  name  is  given  to 
the  perimeter  of  a  circle  ? 

5.  What  is  the  circumference  of  a  circle  whose  radius 
is  35  centimeters  ? 

6.  How  many  millimeters  are  there  in  the  sum  of  the 
circumferences  of  7  nickels  ? 

7.  The  circumference  of  a  circle  whose  diameter  is  35 
millimeters  is  what  fractional  part  of  the  circumference  of 
a  circle  whose  diameter  is  63  millimeters  ? 

8.  The  circumference  of  a  circle  whose  diameter  is  14 
centimeters  is  how  many  times  the  circumference  of  a 
circle  whose  diameter  is  7  centimeters? 

43 


44 


CUMULATIVE  REVIEW. 


9.  Name  the  angle  which  is  the  sum  of 
the  angles  AOB  and  BOC \  the  difference 
of  ADD  and  AOC \  the  difference  of  AOC 
and  ^05. 

10.  Angle  BOC  -f  angle  COB  =  ? 

11.  Angle  A0(7  -h  angle  COD  -  angle  AOB  ^1 

12.  What  sector  is  the  sum  of  the  sec- 
tors FOB  and  DOH^. 

13.  AOB^BOB-BOF^l 

14.  BOH+HOA^{BOF+FOH)  =  '> 

15.  ^0/'-f-^05~(yl02>-yl0^)=? 

16.  How  many  degrees  are  there  in  the 
complement  of  the  angle  formed  by  the  hands  of  a  clock 
at  2  o'clock  ? 

II.  How  many  degrees  are  there  in  the  supplement  of 
the  angle  formed  by  the  hands  of  a  clock  at  4  o'clock  ? 

18.  Make  an  angle  of  60®  whose  sides  are  each  3  in. 
long.  Prolong  the  sides  until  they  are  each  5  in.  long. 
Have  you  increased  the  angle  ? 

Note.  —  An  angle  \s  tlie  amount  of  divergence  of  two  lines  that 
meet  or  that  will  meet  if  sufficiently  prolonged. 

19.  Examine  a  compass.  How  many  degrees  are  in- 
cluded in  the  arc  between  the  points  north  and  southeast  ? 
How  many  between  E.  and  S.E.  ?  Between  N.E.  and 
S.W.?     Between  S.W.  and  N.W.? 

20.  Compare  the  angles  formed  by  the  branches  and 
twigs  of  different  kinds  of  trees. 

21.  Apply  your  protractor  to  the  lines  of  plaid  goods, 
and  find  the  number  of  degrees  in  the  angles  formed  by 
them. 


CUMULATIVE   REVIEW.  45 

22.  Measure  angles  in  patterns  of  wall  paper,  carpets, 
etc.,  until  you  can  estimate  the  number  of  degrees  in  an 
angle  quite  accurately. 

23.  If  we  represent  the  length  of  a  line  by  a;,  how  shall 
we  represent  the  length  of  a  line  twice  as  long  ?  How 
shall  we  represent  the  sum  of  the  lines?  If  x  were  7 
inches,  what  would  3  x^  their  sum,  equal  ?  If  a;  were  10 
inches,  what  would  their  sura  equal?  How  long  is  the 
line  a;  if  3  a;  =  24  inches  ? 

XoTE.  —  A  statement  of  the  equality  of  quantities  is  called  an 
Equation,  as  3  a:  +  5  =  23. 

24.  The  sum  of  two  lines  is  15  inches  and  one  line  is 
twice  the  other.     Find  the  length  of  each. 

Solution.  —  If  we  let  x  equal  the  shorter  line,  2x  would  equal  the 
longer,  and  their  sura  would  be  3  a:.     We  have  the  equations 

a:  +  2x  =  15,  then     a:  =  5,    the  shorter  line, 

then  3  a:  =  15,  then  2  a:  =  10,  the  longer  line. 

25.  Make  an  equation  about  the  length  of  two  lines, 
one  of  which  is  7  times  the  other,  and  their  sum,  40 
inches.  Represent  the  shorter  line  by  x.  What  repre- 
sents the  longer  line  ?  Their  sum  ?  Find  the  length  of 
each  line. 

Note. — Equations  are  used  in  solving  problems  referring  to  any- 
kind  of  units.  Learn  to  use  them  by  putting  the  statement  of  your 
problems  into  equations  whenever  possible,  as  they  will  make  your 
work  much  easier  as  soon  as  you  learn  their  use. 

26.  Make  an  equation,  and  solve  it,  about  the  number 
of  marbles  of  John  and  James.  John  has  5  times  as 
many  as  James  and  the  sum  of  their  marbles  is  18. 

27.  The  sum  of  two  angles  is  150°  and  the  greater  is  4 
times  the  less.     How  many  degrees  are  there  in  each? 

Query.  —  What  shall  x  equal  ? 


x  = 

=  AB 

3x  = 

=  BC 

3x  = 

=  CD 

46  CUMULATIVE   REVTEW. 

28.  The  sum  of  three  lines,  AB,  BC,  and  CD,  is  84 
inches.  BC  and  CD  are  each  3  times  AB.  Find  the 
length  of  each. 

Suggestion.  —  Let 
then 
and 

7x=  84 
from  which  we  find  the  required  values. 

29.  The  sum  of  three  lines,  AB,  CD,  and  EF,  is  81 
inches.     CD  is  twice  EF  and  AB  is  3  times  CD.     Find    / 
the  length  of  each. 

Query.  —  If  x  equals  EF,  what  will  CD  equal?   ABl 

30.  Angle  h  =  twice  angle  a. 
Angle  c  =  6.     How   many  degrees 

has  each  angle,  DF  being  a  straight  >j^^^y 


line?  ^ ""^ 

Note. —  In  an  equation  those  quantities  whose  values  are  known 
are  called  "known  quantities."  In  Ex.  :30  the  known  quantity 
is  180°. 


81.  Angle  h  =  twice  angle  a,  angle 
c  as  3  times  angle  h.  How  many  de- 
grees has  each  angle  ? 

32.   BA  is  perpendicular  to  AC, 
Angle  a  =  angle  x. 
Angle   h  =  twice    angle  a.       How 
many  degrees  has  each  angle  ? 


.lA. 


b 


33.    Angle  a  =  3    times    angle   h. 
Find  the  number  of  degrees  in  each  "^"">:S 

of  the  four  angles. 


cT 


CUMULATIVE  REVIEW.  47 

34.  Angle  h  =  twice  angle  a,  angle  c  =  twice  the  sum 
of  angle  a  and  angle  h.     How  many  de- 
grees has  each  of  the  six  angles  ? 

Query.  —  If  we  know  the  number  of  degrees 
in  angle  a,  how  can  we  teU  the  number  in 
angle/? 

35.  If  we  represent  a  line  by  x^  how  may  we  represent 
a  line  3  inches  longer  ?  If  their  sum  is  15  inches,  what 
equation  would  state  that  fact?  Given  the  equation 
2  a; -1-3  =  15.     Find  the  value  of  x, 

li  2x  and  3  equals  15,  is  it  not  true  that  2x  without 
the  3  equals  15  lacking  3?  If  2  a; =12,  a; =6,  the  shorter 
line,  and  a;+3  =  9,  the  longer  line. 

36.  The   sum  of   the   lines  AB 
and  CD   is   17    inches.      CD  is    5 

inches  longer  than  AB.     Find  the    ^  ^ 

length  of  each. 

Suggestion.  —  Given  the  equation  2  a;  +  5  =  17.  If  we  transpose 
the  5  on  the  left  side  of  the  sign  of  equality  to  the  other  side,  chang- 
ing the  sign,  we  shall  get  the  same  result  as  by  the  reasoning  of  the 
previous  example,  2  a:  =  17  —  5. 

37.  Find  the  value  of  x  when  3  a;  +  7  =  37. 
Find  the  value  of  x  when  4  a;  —  5  =  35. 

Query'.  —  If  4  a;  lacking  5  equals  35,  what  is  the  value  of  4  x  not 
lacking  anything? 

Note.  —  A  quantity  may  be  transposed  from  either  side  of  the 
equation  to  the  other  without  destroying  the  truth  of  the  equation,  if 
the  sign  of  the  quantity  is  changed. 

38.  How  long  is  the  line  x  ii  4:x  —  l  =  Sx-\-5^  the  known 
quantities  referring  to  inches  ? 

Suggestion.  —  Transpose  the  ar's  to  the  left  side,  the  known 
quantities  to  the  right.     When  you  change  the  side,  change  the  sign. 


48  CUMULATIVE  REVIEW. 

39.  Find  the  length  of  the  line  y  if  2y-|-2=y-f  120, 
the  known  quantities  representing  inches. 

40.  Find  the  length  of  the  line  n  if  5n+T  =  4n4-16. 

41.  Find  the  length  of  the  line  a;  if  3a:-7=j:-|-13. 

42.  Find  the  length  of  the  line  a:  if  7a:-8  =  3a;+16. 

43.  If  a  line  8  inches  long  is  represented  by  x^  how 
shall  we  complete  the  following  and  make  a  true  equation  ? 
7ar-10  =  5a;  +  ? 

44.  Complete  8a;  —  3a;  =  ?  x  being  11  centimeters. 
46.    Complete  11a:  —  21  =  ?  x  being  3  decimeters. 

46.  Complete  8a: +  9  4- 8x— 8  =  4a; -H  ?  a;  being  a  line 
4  inches. 

47.  The  sum  of  two  lines  is  61  inches  and  the  greater 
is  5  times  the  less  and  7  inches  more.  Find  the  length 
of  each. 

48.  The  shorter  of  the  two  lines  in  Ex.  47  is  9  inches. 
Substitute  9  for  x  in  the  equation  6  a:  +  7  =  61,  and  see  if 
the  equation  is  true. 

Note.  —  The  process  of  substituting  a  value  for  r  and  showing 
the  equation  to  be  true  is  called  Verification.  It  is  a  convenient  way 
of  proving  the  work. 

49.  The  sum  of  two  angles  is  100"  and  the  greater  is 
40"  more  than  the  less.  Find  each  and  then  verify  your 
work. 

50.  Divide  a  right  angle  into  two  angles,  one  of  which 
is  40"  more  than  the  other. 

51.  How  many  degrees  are  there  in  the  angle  whose  com- 
plement contains  40"  more  than  the  angle  itself  ?     Verify. 


CUMULATIVE  REVIEW.  49 

52.  How  many  degrees  are  there  in  the  angle  whose 
complement  contains  30°  more  than  the  angle  itself  ? 

53.  Divide  an  arc  of  120°  into  two  arcs,  one  of  which 
contains  20°  more  than  the  other. 

54.  A  circular  flower  bed  is  48  feet  in  circumference. 
It  is  bordered  a  part  of  the  way  with  pinks  and  the  rest 
of  the  way  with  mignonette.  The  edge  planted  with 
pinks  is  3  times  as  great  as  that  planted  with  mignon- 
ette. Find  the  number  of  feet  of  the  edge  given  to 
each. 

55.  John  and  James  started  from  the  same  place  and 
ran  round  a  circular  race  track  120  meters  in  circumfer- 
ence until  they  met.  John  ran  20  meters  more  than 
James.     How  far  did  each  run  ? 

Query.  — Can  you  think  how  the  track  and  the  boys  running  on 
it  would  look  ?  It  is  necessary  to  form  clear  pictures  in  your  mind  of 
things  described  in  your  problems. 

56.  John,  James,  and  William  ate  a  round  pie  22  inches 
in  circumference.  The  curved  edge  of  John's  piece  was 
2  inches  longer  than  that  of  James's  and  that  of  James's 
was  3  inches  longer  than  that  of  William's.  Give  the 
length  of  curved  edge  of  each  piece.  What  fractional 
part  of  the  whole  pie  did  each  boy  eat  ? 

57.  The  same  boys  ate  another  round  pie  28  inches  in 
circumference.  William's  piece  was  twice  as  large  as 
James's,  and  James's  was  twice  as  large  as  John's.  Give 
the  length  of  the  curved  edge  of  each  piece. 

58.  Albert,  George,  and  Charles  built  a  fence  around  a 
circular  lot  300  feet  in  circumference.  Albert  built  twice 
as  much  as  George  and  Charles  built  3  times  as  much 
as  George.     How  many  feet  did  each  build  ? 

HORN.    GEOM. — 4 


50  CUMULATIVE  REVIEW. 

69.  Mary,  Jennie,  and  Anna  embroider  the  edge  of  a 
round  tidy  42  inches  in  circumference,  Mary  works  twice 
as  much  as  Jennie,  and  Anna  works  3  times  as  much 
as  Jennie.  How  many  inches  of  the  edge  does  each 
embroider  ? 

60.  Tom,  Fred,  and  Will  whitewashed  the  fence  around 
a  circular  lot  60  feet  in  circumference.  Fred  whitewashed 
3  times  as  much  as  Tom,  and  Will  4  times  as  much  as 
Tom.     How  many  feet  of  fence  did  each  whitewash  ? 

61.  Mary,  Jennie,  and  Anna  decorate  a  round  table 
44  inches  in  circumference.  Mary  decorates  8  inches 
more  of  the  edge  than  Jennie,  and  Anna  twice  as  many 
as  Mary.     How  many  inches  does  each  decorate  ? 

SuooKSTioN.  — 2(x  +  8)=  2x  +  16. 

62.  Divide  a  quadrant  into  two  arcs,  one  of  which  is 
10^  more  than  the  other. 

68.  Three  angles  are  formed  at  the  same  point  on  the 
same  side  of  a  straight  line.  The  first  is  20®  greater  than 
the  second,  and  the  second  is  30°  greater  than  the  third. 
How  many  degrees  has  each  ?     Represent  the  angles. 

64.  Three  angles  are  formed  around  a  common  point. 
The  first  contains  15°  more  than  the  second,  and  the 
second  30°  more  than  the  third.  How  many  degrees  has 
each  ?     Represent. 

65.  A  long  side  of  a  rectangle  whose  perimeter  is  80 
centimeters,  is  5  centimeters  more  than  4 
times  a  short  side.     Find  each  side. 

66.  If  the  angle  a  is  20°  less  than 
twice  the  angle  ft,  how  many  degrees  are 
there  in  each  angle  ? 


CUMULATIVE  REVIEW,  51 


67.  What  are  parallel  lines  ? 

68.  Angle  a  being  107°,  how  many  ^. 
degrees  are  there  in  each  of  the  other 
angles  formed  by  the  transversal  and 
three  parallels  ? 


69.  How  many  degrees  are  there 
in  each  angle  if  angle  c  is  50°  more 
than  angle  e  ? 


70.  How  many  degrees  are  there 
in  each  angle  if  angle  c  is  30°  less 
than  3  times  angle  e  ? 


OF  THE 
OF 


A- 


A- 


CHAPTER   V. 
RECTANGLES. 

NoT£.  —  A  Polygon  in  u  plane  figure  bounded  by  straight  lines. 
When  a  jv)lygon  is  boundeil  by  four  straight  line.s  it  in  called  a 
Quadrilateral.  When  the  o|>|K).site  sides  of  a  quadrilateral  are  par- 
allel it  \h  called  a  Parallelogram.  When  a  parallelogram  has  four 
right  angles  it  is  called  a  Rectangle.  When  the  sides  of  a  rectangle 
are  all  equal  it  is  called  a  Square.  .\  parallelogram  which  is  not 
a  rectangle  is  called  a  Rhomboid. 

1.  Draw  a  polygon.     Is  a  sector  a  polygon  ? 

2.  Draw  a  quadrilateral. 
8.    Draw  a  parallelogram. 

4.  Draw  a  rectangle. 

5.  Draw  a  square  and  show  that  it  is  entitled  to  six 
different  names. 

6.  Draw  a  rhomboid  and  show  to  how  many  names 
it  is  entitled. 

7.  To  how  many  names  is  a  rectangle  entitled  ? 

8.  To  bow  many  names  is  a  triangle  entitled  ? 

9.  Construct  a  rectangle  having  3  rows  of  1-inch 
squares,  each  row  containing  8  squares.  How  many 
square  inches  does  it  cover? 

XoTK.  —  The  amouut  of  surface  which  a  figure  covers  is  called 
its  Area. 


RECTANGLES.  63 

10.  What  is  the  area  of  a  rectangle  consisting  of  2 
rows  of  1-inch  squares,  each  row  containing  4  squares  ? 

11.  What  is  the  area  of  a  rectangle  consisting  of  3 
rows  of  1-inch  squares,  7  squares  in  a  row?  How  long 
is  its  perimeter  ? 

12.  Draw  a  rectangle  whose  base  is  5  inches  and  alti- 
tude 4  inches.     How  many  square  inches  in  area  is  it  ? 

Note.  —  The  lower  base  of  a  rectangle  is  either  one  of  the  sides 
upon  which  it  may  be  supposed  to  stand,  the  upper  base  is  the  side 
opposite  the  lower  base,  and  the  altitude  is  the  distance  between  the 
bases,  measured  perpendicularly. 

13.  In  the  rectangle  ABCD^  what  line  is 
the  lower  base  ?  The  upper  base  ?  What 
lines  represent  the  altitude  ?  If  we  turn  the 
rectangle  and  consider  AC  the  lower  base, 
what  lines  represent  the  altitude  ? 

14.  Show  the  truth  of  the  principle  : 

Principle  16. — The  area  of  a  rectangle  is  equal  to 
the  product  of  its  base  and  altitude. 

15.  What  is  the  area  of  a  rectangle  whose  base  is  12 
inches  and  altitude  half  as  much  ? 

16.  Find  the  area  of  a  rectangle  whose  base  is  5  centi- 
meters and  altitude  8  centimeters. 

17.  Base  =  7|  inches  ;  altitude  =  5  inches  ;  required 
the  area  of  rectangle. 

18.  Base  =  16|  inches  ;  altitude  =  J  of  base  ;  required 
the  area  of  rectangle. 

19.  Base  =  twice  the  altitude ;  3um  of  base  and  alti- 
tude =  15  inches ;  find  the  area  of  rectangle. 

Suggestion.  —  Let  x  =  altitude. 


54  RECTANGLES, 

20.  Base  =  3  times  the  altitude  ;  sum  of  base  and  alti- 
tude =  16  feet ;  find  the  area  of  rectangle. 

21.  How  long  a  fence  will  it  take  to  inclose  a  lot  30 
feet  long  and  20  feet  wide,  and  how  many  square  feet 
will  there  be  in  the  lot  ? 

22.  How  many  square  yards  has  a  rug  12  feet  by  9 
feet,  and  how  many  yards  of  binding  are  required  for  it  ? 

23.  How  many  feet  of  molding  will  be  needed  for  a 
door  frame  8  feet  high  and  4  feet  wide  ? 

QuBRY.  — On  bow  many  sides  is  the  molding  placed? 

24.  If  the  base  of  a  rectangle  is  8  inches,  how  many 
rows  of  square  inches  must  it  have  to  contain  24  square 
inches  ?     What  is  its  altitude  ? 

25.  If  the  base  of  a  rectangle  is  9  inches,  and  its  area 
is  86  square  inches,  how  many  rows  of  squares  has  it  ? 

26.  If  the  base  of  a  rectangle  is  8  inches,  and  its  area 
82  square  inches,  what  is  its  altitude  ? 

27.  Base  of  a  rectangle  =  9  inches  ;  area  =  63  square 
inches  ;  required  the  altitude. 

28.  Base  of  a  rectangle  =  10  centimeters ;  area  =  50 
square  centimeters ;  find  tlie  altitude. 

29.  Base  =  10  inches ;  area  =  35  square  inches;  re- 
quired the  altitude. 

30.  Base  =  12J  inches  ;  area  =  50  square  inches  ;  re- 
quired the  altitude. 

81.  Base  =  7  J  inches  ;  area  =  60  square  inches  ;  re- 
quired the  altitude. 

32.  Base  =  4J  inches  ;  area  =  94J  square  inches  ;  re- 
quired the  altitude. 


RECTANGLES.  55 

33.  Base  =  8  inches  ;  area  =  72  square  inches  ;  re- 
quired the  altitude  and  perimeter. 

34.  Base  =  9  centimeters  ;  area  =  45  square  centi- 
meter ;  find  the  altitude  and  perimeter. 

35.  How  many  millimeters  are  there  in  the  perimeter  of 
a  rectangle  whose  base  is  21  millimeters  and  area  105 
square  millimeters  ? 

36.  How  many  millimeters  are  there  in  the  perimeter 
of  a  rectangle  whose  base  is  3  centimeters  and  whose  area 
is  6  square  centimeters? 

37.  How  many  yards  of  binding  are  required  for  an 
oilcloth  which  is  10  feet  long  and  contains  90  square  feet  ? 

38.  How  many  yards  of  binding  are  required  for  a  rug 
15  feet  long  which  contains  20  square  yards  ? 

39.  How  many  yards  of  fringe  are  required  for  the 
ends  of  a  pair  of  curtains,  each  of  which  is  10  feet  long, 
and  contains  50  square  feet  ? 

40.  If  the  altitude  of  a  rectangle  is  3  inches,  and  it 
contains  30  square  inches,  how  many  square  inches  must 
there  be  in  each  row  ?    What  is  the  base  of  the  rectangle  ? 

41.  If  the  area  of  a  rectangle  is  16  square  inches,  and 
its  altitude  is  2  inches,  how  many  square  inches  are  there 
in  each  row  ? 

42.  If  the  altitude  of  a  rectangle  is  7  inches,  and  its 
area  is  56  square  inches,  what  is  its  base  ? 

43.  Altitude  of  rectangle  =  8  inches  ;  area  =  96  square 
inches  ;  find  base. 

44.  Altitude  of  rectangle  =  9^  inches ;  area  =  95  square 
inches  ;  find  base. 


66  RECTANGLES, 

45.  Altitude  of  rectangle  =  6J  inches  ;  area  =  50 
square  inches  ;  find  base. 

46.  Altitude  of  rectangle  =  4J  inches  ;  area  =  83J 
square  inches  ;  find  base. 

47.  Altitude  of  rectangle  =  8.5  inches;  urea  =  170 
square  inches  ;  find  base. 

48.  Altitude  of  rectangle  =  7.5  inches;  area  =  87.5 
square  inches  ;  find  base. 

49.  Altitude  of  rectangle  =  24  millimeters  ;  area  = 
288  square  millimeters;  find  base. 

50.  Altitude  of  rectangle  =  3^  centimeters ;  area  =  77 
square  centimeters  ;  find  base. 

51.  How  many  centimeters  are  there  in  the  perimeter 
of  a  rectangle  whose  altitude  is  15  centimeters  aiul  area 
225  square  centimeters  ? 

52.  How  many  millimeters  are  there  in  the  perimeter 
of  a  rectangle  whose  area  is  15  square  centimeters  and 
whose  altitude  is  5  centimeters? 

53.  If  the  altitude  of  a  rectangle  is  9  inches  and  its 
area  is  81  square  inches,  how  long  is  the  base?  What 
kind  of  a  rectangle  is  it? 

54.  What  is  one  side  of  a  square  whose  area  is  9  square 
inches  ? 

55.  Find  the  perimeter  and  the  area  of  a  square  whose 
side  is  5  inches. 

56.  Find  the  perimeter  of  a  square  whose  area  is  36. 
square  inches  ;  of  one  whose  area  is  49  square  inches ;  64 
square  inches ;  81  square  inches ;  100  square  inches. 


RECTANGLES,  §7 

57.  How  many  square  inches  are  there  in  two  rectan- 
gles, the  first  being  20  inches  long  and  10  inches  wide, 
the  other  19  inches  long  and  11  inches  wide  ? 

58.  How  many  square  inches  are  there  in  three  squares, 
the  side  of  the  first  being  4  inches,  of  the  second  6  inches, 
of  the  third  8  inches  ? 

59.  How  many  square  inches  are  there  in  two  squares, 
the  perimeter  of  the  first  being  72  inches,  and  that  of  the 
second  40  inches  ? 

60.  Find  the  combined  length  of  the  perimeters  of  two 
squares,  one  of  which  contains  36  square  inches,  and  the 
other  25  square  inches. 

61.  How  many  rods  of  fencing  will  be  required  to 
inclose  two  square  fields,  one  of  which  contains  9  square 
rods,  the  other  49  square  rods? 

62.  Draw  a  figure  which  is  the  sum  of  two  squares  placed 
so  that  they  have  a  common  side,  each  square  being  7 
inches  in  length  and  breadth.    Find  its  area  and  perimeter. 

63.  Draw  a  figure  which  is  the  sum  of 
two  squares  placed  together  on  a  line,  one 
5  inches,  the  other  4  inches  in  dimensions, 
and  find  its  area  and  perimeter. 

64.  Draw  a  figure  which  is  the  sum  of 
three  squares  placed  together  on  a  line, 
one  6  inches,  another  5  inches,  the  other 
4  inches  in  dimensions,  and  find  its  area 
and  perimeter. 

Qb,  Find  the  difference  in  the  area  of  two  rectangles, 
one  13  inches  long  and  11  inches  wide,  and  the  other  11 
inches  long  and  5  inches  wide. 


58 


RECTANGLES. 


66.  Find  the  difference  in  the  area  of  two  squares 
whose  sides  are  respectively  7  inches  and  5  inches. 

67.  Find  the  difference  in  area  between  two  squares, 
the  i>erimeter  of  the  greater  being  32  inches,  and  that  of 
the  less  being  28  inches. 

68.  Find  the  difference  in  the  number  of  rods  of  fence 
required  to  inclose  two  square  fields,  one  containing  64 
square  rods,  the  other  36  square  rods. 

69.  Cut  a  4-inch  square  from  the  upper 
right-hand  corner  of  a  7-inch  square,  and  find 
the  area  and  {perimeter  of  the  figure  remaining. 

70.  Draw  a  figure  which  is  the  difference 
of  two  squares,  one  6  inches  and  the  other  4 
inches  in  dimensions,  the  smaller  square  being  cut  from 
the  right-hand  lower  comer  of  the  larger,  and  find  its  area 
and  iHjrimeter. 

71.  Draw  a  figure  which  is  the  difference  of  two 
squares,  one  8  inches  and  the  other  5  inches  in  dimen- 
sions, the  smaller  being  subtracted  at  the  left-hand  lower 
corner  from  the  larger,  and  find  it8  area  and  perimeter. 

72.  Place  a  square  3  inches  in  dimensions  in  the  upper 
left-hand  corner  of  one  10  inches  in  dimensions,  and  find 
the  area  and  jjerimeter  of  the  figure  which  shows  their 
difference. 

73.  Give  the  length,  width,  and  area 
of  the  rectangle  which  must  be  cut  off 
from  the  given  rectangle  that  the  largest 
possible  square  may  remain. 

74.  Give  the  dimensions  and  area  of 
the  rectangle  which,  added  to  one  side  of 
the  given  rectangle,  will  make  it  a  square. 


tin. 


RECTANGLES.  59 

75.  Give  the  number  of  square  inches  in  the  two  rec- 
tangles and  little  square  which,  cut  off  from  the 
given  8-inch  square,  will  reduce  it  to  a  7 -inch 
square. 

76.  Find   the   number  of   square    inches   in 
the  two  rectangles  and  little  square  which,   added  to  a 
5-inch  square,  will  make  it  a  7 -inch  square. 

77.  If  a  5-inch  square  is  cut  off  from  the  right-hand 
upper  corner  of  a  10-inch  square,  how  many  square  inches 
will  there  be  in  the  irregular  figure  remaining  ?  Find  its 
perimeter. 

78.  If  a  4-inch  square  is  cut  from  the  lower  left-hand 
corner  of  a  9-inch  square,  what  is  the  area  and  what  is 
the  perimeter  of  the  remaining  figure  ? 

79.  Add  to  a  5-inch  square  the  two  rectangles  and 
small  square  that  will  make  it  a  6-inch  square,  and  find 
the  area  of  the  additions. 

80.  If  a  2-inch  square  is  added  at  the 
right-hand  upper  corner  of  a  7-inch  square, 
what  is  the  area  of  the  irregular  figure  thus 
formed  ?  How  long  is  its  perimeter  ?  How 
many  sides  has  it  ? 

Note.  —  A  plane  figure  bounded  by  six  sides  is  called  a  Hexagon. 
The  figure  in  Ex.  80  represents  a  certain  kind  of  hexagon. 

81.  Draw  several  differently  shaped  figures  each  hav- 
ing six  sides  and  write  upon  them  the  name  which  belongs 
to  them 

82.  Find  the  area  of  the  figure  which  is  the  sum  of 
two  squares,  respectively,  9  inches  and  6  inches  in  dimen- 
sions, so  placed  that  a  side  of  one  square  is  a  continuation 


00  HECTAXGLES. 

of  a  side  of  the  other,  and  an  angle  of  one  square  is  adja 
cent  to  an  angle  of  the  other.     Find  its  perimeter. 

.    83.    How  many  square  inches  are  there  in     ^ ^ 

the  figure  (represented  by  ABCDEF)  which    ^ 

is  the   difference  of   the   square  ODCB^  10 

inches  eacli  way,  and  QEFA^  7  inches  each 

way?  ^'         ^    *- 

84.  Draw  the  square  of  the  line  AB^  5  inches  long. 

NotE.  —  The  square  of  a  line  is  the  square  of  which  the  line 
forms  a  side. 

85.  Draw  the  recUmgle  of  the  line,  AB  and  BC^  AB 
being  8  inches,  BC^  7  inches. 

Note.  —  The  rectanf^le  of  two  lines  \»  the  right-angled,  four-eided 
plane  figure  formed  by  uxing  one  of  the  lines  as  the  measure  of  length 
and  the  other  as  the  measure  of  width.  It  is  also  called  the  product 
of  the  lines. 

8(^.  Draw  a  figure  whirli  is  tlit*  sum  of  the  squares  of 
two  lines,  each  (>  inches,  and  determine  the  number  of 
s({uare  inches  in  it. 

87.  Place  the  squares  of  two  lines,  each  7  inches,  so 
that  they  shall  have  one  common  side,  and  find  the  area 
and  i>erimeter  of  the  rectangle  thus  formed. 

88.  Find  the  perimeter  of  the  rectangle  which  is  the 
sum  of  the  squares  of  two  lines,  each  6  inches,  so  placed 
that  they  have  a  common  side. 

89.  Place  the  squares  of  two  lines,  each  8 
inches,  so  that  the  vertex  of  one  of  the  angles 
of  each  square  shall  coincide  with  the  middle 
point  of  a  side  of  the  otlier,  and  find  the 
perimeter  of  the  figure  thus  formed.  How  many  sides 
has  the  figure  ? 


RECTANGLES,  61 

90.  Place  the  squares  of  two  lines,  each  10  inches,  so 
that  one  half  of  a  side  of  each  square  shall  coincide  with 
one  half  of  a  side  of  the  other,  and  find  the  perimeter  of 
the  figure  thus  formed. 

91.  Find  the  area  and  perimeter  of  the  figure  formed 
by  subtracting  the  square  of  a  2-inch  line  from  tlie  upper 
left-hand  corner  of  the  square  of  a  7 -inch  line. 

92.  Find  the  area  and  the  perimeter  of  the  figure  which 
is  the  difference  of  the  squares  of  the  lines  AB^  %  inches, 
and  A  (7,  6  inches,  the  subtraction  being  made  from  the 
upper  right-hand  corner. 

93.  Place  the  square  of  the  line  AB^  6 
inches,  inside  the  square  of  the  line  (7i>,  10 
inches,  so  that  the  middle  points  of  their 
bases  shall  coincide. 

How  many  square  inches  are  there  in  the  figure  which 
is  the  difference  of  their  squares  ?  How  long  is  its  perim- 
eter ?     How  many  sides  has  it  ? 

94.  Find  the  area  and  perimeter  of  the  figure  which 
is  the  difference  of  the  squares  of  the  lines  AB^  9  inches, 
and  (7i>,  7  inches,  placed  as  in  Ex.  93. 

95.  A  mantel  40  inches  high  and  equally 
wide  is  set  with  a  grate  23  inches  high  and 
wide.  How  many  square  inches  are  there 
in  the  mantel  ? 


96.  Place  the  small  square  in  Ex.  93,  one  inch  farther 
to  the  left,  and  find  the  area  and  the  perimeter  of  the 
figure  which  is  the  difference  of  the  squares. 

97.  Place  the  same  squares  so  that  their  middle  points 
coincide.  How  many  square  inches  are  there  in  their 
difference  ? 


62  RECTANGLES. 

98.  Cut  a  square  3  inches  in  dimensions  from  the 
middle  of  a  square  7  inches  in  dimensions,  and  tell  how 
many  square  inches  are  left. 

99.  The  frame  of  a  looking-glass  is  30  inches  in  length 
and  width  ;  the  glass  is  20  inches  square.  How  many 
square  inches  are  there  in  the  frame? 

100.  A  garden  40  feet  square  gives  so  mjicli 
space  to  a  walk  along  its  edge  that  the  remain- 
ing space  is  28  feet  square.  How  many  squan' 
feet  are  there  in  the  walk,  and  what  is  its 
width  ? 

101.  The  perimeter  of  a  certain  square  is  4  inches 
longer  than  that  of  a  smaller  square,  and  the  sum  of  their 
perimeters  is  36  inches.  How  many  square  inches  are 
there  in  the  sum  of  the  squares? 

SuoQESTiON.  —  Let  X  =  perimeter  of  smaller  square. 

102.  The  perimeter  of  a  square  is  8  inches  longer  than 
that  of  a  smaller  square,  and  the  sum  of  their  perimeters 
is  56  inches.  How  many  square  inches  are  there  in  the 
difference  of  their  squares  ? 

103.  Draw  a  diagram  of  a  rectangular  lot  7  rods  long 
and  5  rods  wide,  representing  each  rod  by  a  lialf  inch, 
and  show  its  area. 

104.  Draw  a  diagram  of  a  room  7  feet  long  and  5  feet 
wide,  representing  each  foot  by  a  centimeter,  and  find  the 
area  of  the  floor. 

105.  Draw  a  diagram  of  a  rug  12  feet  long  and  9  feet 
wide  on  a  scale  of  1  centimeter  to  the  foot. 

106.  The  bottom  and  each  of  the  sides  of  a  drawer  12 
inches  long,  10  inches  wide,  and  4  inches  deep,  are  in  the 


RECTANGLES.  .  63 

shape  of   what  geometrical  figure  ?     How  many  square 
inches  of  blue  velvet  would  it  take  to  line  the  drawer? 

107.  How  many  square  inches  are  there  in  all  the  sur- 
faces of  a  box  8  inches  long,  6  inches  wide,  4  inches  high  ? 
How  many  pairs  of  equal  rectangular  faces  has  the  box  ? 

108.  How  many  square  inches  are  there  in  all 
the  surfaces  of  a  brick  8  inches  long,  4  inches 
wide,  and  2  inches  thick  ?  How  many  pairs  of 
equal  rectangular  faces  has  the  brick  ? 

Note.  —  A  solid  which  has  six  rectangular  faces,  of 
which  the  opposite  ones  are  equal  and  parallel,  is  called  a  Right 
Prism  or  Parallelopiped. 

109.  Is  the  space  inside  the  drawer  mentioned  in  Ex. 
106  a  geometric  or  a  physical  solid  ? 

Note.  —  Any  material  object  is  a  physical  solid,  as  a  book,  or  box, 
or  house.  The  space  occupied  by  a  material  object  forms  a  geometric 
solid. 

110.  Think  of  a  chalk  box  and  imagine  the  geometric 
solid  which  corresponds  to  it.  What  name  is  given  to 
that  geometric  solid  ? 

111.  Were  you  ever  inside  of  a  parallelopiped  ? 

112.  How  many  square  feet  are  there  in  the  walls,  ceil- 
ing, and  floor  of  a  room  10  feet  long,  9  feet  wide,  8  feet 
high? 

113.  What  name  is  given  to  those  prisms  _^ 
whose  faces  are  all  squares  ?                               ll^^^^^lll 

114.  How  many  square  inches  are  there    I       I  ||l 

in  all  the  faces  of  a  cube,  one  edge  of  which     ||;W'^'-"--"l^ 
is  5  inches  ? 

115.  How  many  inches  are  there  in  all  the  lines  formed 
by  the  meeting  of  two  surfaces  of  the  same  cube  ? 


64  .  RECTANGLES. 

116.  How  many  angles  are  formed  on  each  face  of  a 
cube  by  its  boundary  lines? 

117.  Find  the  number  of  degrees  in  the  sum  of  all  the 
angles  formed  on  the  faces  of  a  cube  by  their  boundary 
lines. 

118.  How  many  square  inches  are  there  in  the  surfaces 
of  a  box  whose  height  is  4  inches,  its  width  twice  its 
height,  and  its  length  3  times  its  width? 

119.  How  many  square  inches  are  there  in  the  surfaces 
of  a  brick  which  is  0  inches  long,  \  as  wide  as  long,  and 
\  as  thick  as  wide  ? 


CHAPTER  VI. 

TRIANGLES   AND   LINES. 

1.  Draw  a  line  5  inches  long.  With  one 
end  of  the  line  as  a  center  and  a  radius  of  5 
inches,  describe  an  arc.  With  the  other  end 
as  a  center  and  the  same  radius,  describe  an 
arc  intersecting  the  first.  Join  the  point  of 
intersection  with  the  ends  of  the  line.  What  kind  of  a 
plane  figure  is  thus  formed  ? 

Note.  —  A  triangle  whose  sides  are  all  equal  is  called  an  Equi- 
lateral Triangle. 

2.  How  many  centimeters  are  there  in  the  perimeter 
of  an  equilateral  triangle,  one  side  of  which  is  8  milli- 
meters ? 

3.  Find  the  length  in  decimeters  of  one  side  of  an 
equilateral  triangle  whose  perimeter  is  48  centimeters. 

4.  Find  the  difference  between  the  length  of  the  perim- 
eter of  an  equilateral  triangle  each  of  whose  sides  is  7 
inches,  and  that  of  an  equilateral  rectangle  each  of  whose 
sides  is  7  inches. 

Query.  —  What  does  "equilateral"  mean? 

5.  The  perimeter  of  a  triangle  whose  sides  are  each  6 
centimeters  is  what  fractional  part  of  the  perimeter  of  a 
hexagon  each  of  whose  sides  is  6  centimeters  ? 

HORN.   GEOM. 5  65 


66  TRIANGLES  AND  LINES. 

6.  Draw  two  equal  lines  making  any  angle,  and  join 
their  extremities.     What  plane  figure  is  formed  ? 

Note.  —  A  triangle  which  has  two  of  its  sides  equal  is  called  an 
Isosceles  Triangle.    The  unequal  side  is  called  the  Base. 

7.  Find  the  base  of  an  isosceles  triangle  whose  perim- 
eter is  90  millimeters  and  each  of  whose  equal  sides  is 
35  millimeters. 

8.  Find  ejvch  of  the  equal  sides  of  an  isosceles  tri- 
angle whose  perimeter  is  40  inches  and  base  10  inches. 

9.  Each  of  the  equal  sides  of  an  isosceles  triangle  is 
double  the  base,  and  the  perimeter  is  45  centimeters. 
Find  each  side. 

SuQOESTioN.  —  I^t  X  =  base. 

10.  The  base  of  an  isosceles  triangle  is  5  inches  longer 
than  each  of  its  equal  sides,  and  the  perimeter  is  35  inches. 
Find  each  side. 

Query.  —  What  shall  x  equal  ? 

11.  The  sum  of  the  equal  sides  of  an  isosceles  triangle 
is  4  times  the  base,  and  the  perimeter  is  15  inches.  Find 
each  side. 

12.  Cut  out  an  isosceles  triangle,  fold  it  so  that  the 
equal  sides  shall  coincide,  and  show  the  truth  of  the  fol- 
lowing theorem: 

Principle  17. — In  an  isosceles  triangle  the  angles 
opposite  tlie  equal  sides  are  equal. 

13.  In  the  isosceles  triangle  ABO 
the  angle  BAC  is  70°.     How  many 

degrees   are   there   in    each    exterior     

angle  formed  by  prolonging  the  side 
ACi     Quote  principle. 


TRIANGLES  AND  LINES.  67 

Note.  —  An  angle  formed  by  prolonging  one  side  of  a  polygon  is 
called  an  Exterior  Angle. 

14.  The  exterior  angle  BBQ^  formed  by 
prolonging  one  leg  of  the  isosceles  triangle 
ABO^  is  115°.  Find  each  of  the  base  angles 
and  the  exterior  angle  BCE. 

15.  The   exterior   angle   formed  by  pro- 
longing the  base  of  an  isosceles  triangle  con- 
tains 3  times  as  many  degrees  as  the  interior  base  angle. 
How  many  degrees   are  there   in   the  sum  of   the   base 
angles  ?     How  many  in  their  difference  ? 

16.  The  exterior  angle  formed  by  prolonging  one  of 
the  legs  of  an  isosceles  triangle  is  26°  more  than  a  base 
angle.     How  many  degrees  are  there  in  each  base  angle  ? 

17.  Draw  and  cut  out  an  equilateral  triangle.  Fold  it 
so  that  two  of  its  equal  sides  shall  coincide.  Are  the 
angles  opposite  them  equal  ?  Smooth  it  out  and  fold  it  so 
that  another  pair  of  equal  sides  shall  coincide.  Are  the 
angles  all  equal  ? 

Note.  —  A  triangle  whose  angles  are  all  equal  is  Equiangular. 

Principle  18. — An  equilateral  triangle  is  equi- 
angular. 

18.  In  the  equilateral  triangle  ABO^  of  which  BC  is 
the  base,  which  is  the  greater,  the  vertical  angle  or  one  of 
the  base  angles  ? 

Note.  —  Any  side  of  a  triangle  on  which  it  may  be  supposed  to 
stand  may  be  called  the  Base,  and  the  angle  opposite  the  base  is  called 
the  Vertical  Angle. 

19.  If  AB  is  the  base  of  the  triangle 
ABC,  what  is  the  vertical  angle  ?  If  ^  is 
the  vertical  angle,  what  is  the  base  ? 


68  TRIANGLES  AND  LINES. 

20.  Draw  three  equilateral  triaugles  uf  the  same  dimen- 
sions and  place  them  so  that  they  have  the  vertex  of  an 
angle  of  each  at  a  common  point,  as  at 
C  in  the  figure.  It  will  be  seen  that  a 
straight  line  is  formed  by  the  bases  of 
the  outer  triangles. 

The  straight  line  AOE  is  what  frac- 
tional part  of  the  broken  line  ABLE  ? 

21.  How  many  degrees  are  there  in  the  sum  of  the 
angles  at  C'i     Quote  geometric  principles. 

22.  How  many  degrees  are  there  in  each  of  the  angles 
at  (7?     Give  reasons. 

28.  How  many  degrees  are  there  in  angle  BAO?  In 
angle  ABC?  How  many  are  there  in  each  angle  of  each 
equilateral  triangle  ?  How  many  in  the  sum  of  the 
angles  of  either  one  of  the  equilateral  triangles  ? 

24.  How  many  degrees  are  there  in 
the  angle  x,  formed  by  prolonging  a  side 
of  an  equilateral  triangle  ? 

25.  Draw  an  equilateral  triangle  and  prolong  its  sides 
so  as  to  form  an  exterior  angle  at  each  vertex.  Cut  out 
the  exterior  angles  and  place  them  around  a  common 
point.  Will  they  form  a  continuous  surface  around  the 
point  ?     How  many  degrees  are  there  in  their  sum  ? 

26.  Draw  a  regular  polygon  of  three  sides  and  name  it. 

NoTK.  —  When  a  polygon  is  equilateral  and  equiangular  it  is  called 
a  Regular  Polygon. 

27.  Draw  a  regular  polygon  of  four  sides  and  write  its 
name  upon  it.  How  many  right  angles  has  it?  How 
many  degrees  are  there  in  all  its  angles  ? 


TRIANGLES  AND  LINES.  69 

28.  Cut  out  four  equal  squares  and  place  them  around 
a  common  point  so  that  an  angle  of  each  shall  have  its 
vertex  at  the  point.  Will  they  form  a  continuous  surface 
around  the  point  ? 

29.  Make  six  equal  equilateral  triangles  and  place  them 
around  a  common  point,  as  E.  They 
form  a  regular  hexagon.  How  many 
degrees  are  there  in  each  angle  formed 
at  U?  How  many  degrees  are  there  jl(^  Y^  V 
in  angle  ABC?  BCD?  In  each  angle 
of  the  hexagon  ?     Give  reasons. 

30.  How  many  sides  has  the  poly- 
gon ADCB  ?     Which  of  its  sides  are  parallel  ? 

Note.  —  A  quadrilateral  which  has  only  two  of  its  sides  parallel 
is  called  a  Trapezoid. 

31.  Draw  a  trapezoid  and  show  into  how  many  trian- 
gles it  can  be  divided  by  one  line. 

32.  If  each  of  the  sides  of  the  equilateral  triangles  in 
Ex.  29  is  8  centimeters,  what  is  the  perimeter  of  the 
trapezoid  ^6^i>  6^? 

33.  Inclose  a  surface  by  three  unequal  lines,  and  name 
the  plane  figure  thus  formed. 

Notp:. —  A  triangle  whose  sides  are  all  unequal  is  called  a  Scalene 
Triangle. 

34.  Construct    a   triangle   whose   sides   are   5   inches, 

6  inches,  and  7  inches,  using  the  7-inch  line  as  the  base. 
Suggestion.  —  With    the   extremities    of    the   base    as    centers 

describe  arcs,  with  radii  respectively  5  inches  and  6  inches,  and  join 
the  point  of  intersection  with  the  extremities  of  the  base. 

35.  Construct  a  triangle  whose  sides  are  8  centimeters, 

7  centimeters,  and  6  centimeters. 


70  TRIANGLES  AND   LINES. 

36.  Construct  a  triangle  whose  sides  are  9  inches,  3 
inches,  and  4  inches.     Explain. 

37.  What  is  the  perimeter  of  the  scalene  triangle  ABC^ 
in  whicli  AB  is  12  inches,  EC  \&  2  inches  longer  than  AB^ 
and  AC'\%^  inches  longer  than  BCi 

38.  The  perimeter  of  a  scalene  triangle  is  47  inches; 
one  side  is  11  inches  and  another  side  is  1}  times  as  long. 
Find  the  third  side. 

39.  The  triangle  ABC^  whose  perimeter  is  54  inches, 
has  the  side  AB  7  inches  longer  than  the  side  BC^  and 
the  side  BC  10  inches  longer  than  the  side  AC,  Find 
each  side. 

Suggestion.  —  Let  x  —  AC. 

40.  The  scalene  triangle  ABC  has  the  side  AB  12 
inches  longer  tlian  the  side  ACy  and  the  side  AC  ^  inches 
longer  tlian  the  side  BC,  The  perimeter  is  73  inches. 
Find  each  side. 

41.  The  side  XY  of  the  scalene  triangle  XYZ  is  11 
inches  longer  than  the  side  YZ,  and  the  side  XZ  is  17 
inches  longer  than  the  side  YZ,  The  perimeter  is  88  inches. 
Find  each  side. 

42.  The  side  DE  of  the  scalene  triangle  DEF^  the  per- 
imeter of  which  is  65  inches,  lacks  8  inches  of  being  twice 
as  long  as  the  side  EF.  The  side  DF  lacks  17  inches  of 
being  3  times  as  long  as  EF,  Find  the  length  of  each 
side. 

43.  Quote  the  geometric  principle  which  tells  how 
many  degrees  there  are  in  the  sum 

of  the  angles  a,  6,  and  c,  AB  being  \      / 

a  straight  line.  a i^ s 


TRIANGLES  AND  LINES.  71 


44.  Draw  and  cut  out  a  triangle, 
dividing  each  of  its  angles  by 
straight  lines.  Cut  off  two  of  the 
corners.  Place  them  beside  the 
third  corner. 

It  will  be  seen  that  all  the  angles 
will  have  their  vertex  at  a  common 
point,  and  will  lie  on  the  same  side 
of  a  straight  line.  How  many 
degrees  are  there  in  the  sum  of  all 
the  angles? 


Principle  19.  —  The  sum  of  the  angles  of  any  triangle 
is  equal  to  two  right  angles,  or  180  degrees. 

45.  Draw  different  kinds  of  triangles,  and  repeat  the 
process  given  in  Ex.  44  illustrating  Prin.  19. 

46.  How  many  degrees  are  there  in  the  vertical  angle 
of  an  isosceles  triangle  whose  base  angles  are  each  80°  ? 

47.  How  many  degrees  are  there  in  each  base  angle  of 
an  isosceles  triangle  whose  vertical  angle  is  50°  ? 

48.  The  angle  a  of  a  triangle  is  80°,  and  angle  5  is  3 
times  angle  c.     Find  angle  h  and  angle  c.     Let  x  =  1 

49.  Make  a  right  angle,  and  join  the  extremities  of  the 
lines.  How  many  degrees  are  there  in  the  sum  of  the 
two  angles  that  are  not  right  angles  ? 

Note.  —  A  triangle  which  has  a  right  angle  is  a  Right  Triangle. 
The  side  opposite  the  right  angle  is  called  the  Hypotenuse. 

50.  How  many  degrees  are  there  in  each  angle  of  an 
isosceles  triangle  whose  vertical  angle  is  equal  to  the  sum 
of  the  base  angles  ? 


72  TRIANGLES  AND  LINES. 

51.  Draw  a  right-angled  isosceles  triangle  having  its 
equal  sides  each  5  inches  long,  and  show  how  many 
degrees  tliere  are  in  each  of  the  complementary  angles. 

Query. —  When  is  one  angle  complementary  to  another? 

52.  Draw  a  right-angled  isosceles  triangle  having  its 
equal  sides  each  10  inches  long,  and  show  how  many 
degrees  there  are  in  each  of  the  complementary  angles. 

53.  In  the  scalene  triangle  ABC  the  angle  AisA  right 
angle,  and  the  angle  ^  is  4  times  the  angle  C,  How 
many  degrees  are  there  in  each  of  the  complementary 
angles  ? 

54.  Find  each  of  the  complementary  angles  in  a  right 
triangle,  in  which  one  acute  angle  is  5  times  the  other. 

55.  Construct  an  angle  of  60®  at  one  extremity  of  a 
line  and  one  of  70®  at  the  other  extremity.  Prolong  the 
lines  until  they  meet.  How  many  degrees  are  there  in 
the  third  angle  ? 

Note.  —  A  triangle  in  which  all  the  angles  are  acute  is  an  Acute- 
angled  Triangle. 

56.  Draw  an  acute-angled  triangle  in  which  one  angle 
is  80®. 

57.  Draw  an  obtuse  angle,  and  join  the  ends  of  the 
lines  which  form  it.  What  kind  of  triangle  is  thus 
formed  ? 

Note.  —  A  triangle  which  has  an  obtuse  angle  is  an  Obtuse-angled 
Triangle. 

58.  Can  you  draw  an  acute-angled  triangle  in  which 
the  sum  of  any  two  angles  is  less  than  a  riglit  angle  ? 

59.  To  what  is  the  sum  of  the  oblique  angles  of  a  right 
triangle  equal  ? 


TRIANGLES  AND  LINES. 


73 


60.  Try  to  draw  an  obtuse-angled  triangle  in  which 
the  sum  of  the  acute  angles  is  greater  than  a  right  angle. 
Explain. 

61.  JBB  is  perpendicular  to  AC,  one  of 
the  legs  of  the  isosceles  triangle  ABC, 
whose  vertical  angle  is  40°.  How  many 
degrees  are  there  in  each  of  the  angles  x, 
«/,  and  z  2 

62.  BD  bisects  the  vertical  angle  of  the 
isosceles  triangle  ABC,  a  base  angle  of 
which  is  Q3°.  How  many  degrees  are 
there  in  angle  x?  In  angle  y?  In  angle 
m?  In  angle  k?  What  is  the  relative 
position  of  BI)  and  AC? 

63.  AD  is  a  bisector  of  a  base  angle  of 
the  isosceles  triangle  whose  vertical  angle 
B  is  48°.  How  many  degrees  are  there 
in  angle  ^i)^? 

64.  In  the  isosceles  triangle  ABC  the 
vertical  angle  is  38°.  AU  bisects  one  base 
angle.  DC  bisects  the  other.  Angles  of 
how  many  degrees  are  formed  at  F? 

65.  Angles  of  how  many  degrees  are 
formed  by  the  intersection  of  the  bisector  of  the  vertical 
angle  and  the  bisector  of  a  base  angle  in  the  triangle 
whose  base  angles  are  each  20°  ? 

66.  What  angles  are  formed  by  the  intersection  of  the 
lines  drawn  from  the  vertices  of  the  base  angles  perpen- 
dicular to  the  legs  of  an  isosceles  triangle  whose  vertical 
angle  is  30°? 


74 


TRIANGLES  AND  LINES, 


67.  Angle  a  =  55*.  Angle  h  =  60**. 
Angle  c  =  ?  Angle  c?  =  ?  Which  is  the 
greater,  a  +  6,  or  (^  ? 

68.  Exterior  angle  ACB  =  twice 
angle  ACB  \  ABC  =10'';  find  angle 
BAC.  Which  is  the  greater,  the  ex- 
terior angle,  or  the  sum  of  tlie  two  in- 
terior angles  which  are  opposite? 

69.  BEF  is  an  equilateral  triangle. 
How  many  degrees  has  the  exterior  angle 
DEG  ?  Compare  tlie  number  of  degrees 
in  the  exterior  angle  with  the  number 
of  degrees  in  the  opposite  interior  angles. 


70.  Draw  an  isosceles  triangle  whose  vertical  angle  is 
30®,  and  compare  the  number  of  degrees  in  any  exterior 
angle  with  the  number  of  degrees  in  the  opposite  interior 
angles.  Can  you  give  a  reason  for  the  fact  stated  in  the 
following  theorem  ? 

Principle  20. — jin  exterior  angle  of  a  triangle  is 
equal  to  the  sum  of  the  opposite  interior  angles. 

71.  Find  angle  c  and  angle  6. 

72.  Every  degree  is  divided  into  60  equal 
parts,  called  minutes,  marked  (').  If  one  angle 
of  a  triangle  is  37°  30',  and  another  is  50°  30',  how 
many  degrees  are  there  in  the  angle  exterior  to  the  third 
angle  ? 

73.  How  many  degrees  are  there  in  the  exterior  ver- 
tical angle  of  a  triangle,  each  of  whose  base  angles  is 
27°  15'? 


TRIANGLES  AND  LINES. 


15 


74.  How  many  degrees  are  there  in  the  exterior  vertical 
angle  of  an  isosceles  triangle,  each  of  whose  exterior  base 
angles  is  110°? 

75.  Angle  a;  =  55°.     Angle   z  =  ? 
Angle  «/  =  48°.     Angle  w  =  ? 

76.  What  angles  are  exterior  to  the  tri- 
angle whose  angles  are  g,  /,  and  d  ?  Name 
the  angle  exterior  to  the  triangle  whose 
angles  are  g,  h,  and  k?  If  a  =  120°,  c  =  20°, 
/=91°,  and  h  =  65°,  how  many  degrees  are 
there  in  each  angle  of  the  three  triangles  ? 


77.    Given,  angle   m  =  35°,  angles  =  30° 
angle  e  =  40°,  angle  ^  =  40°,  angle  5  =  20°. 
Required,  the  angles  Z,  g,  A,  A;,  c?,  a,  /. 


78.  What  angle  is  exterior  to  the 
triangle  whose  angles  are  e, .  c?,  and 
/  ?  ^,  A,  and  k?  a,  b,  and  c?  m,  I, 
and  n? 


79.  Try  to  draw  an  isosceles  triangle  having  an  exterior 
base  angle  of  75°.     Explain. 

80.  Let  AB  be  a  given  line  and  C  a  point  above  it. 
DraAV  the  perpendicular  CD  and  the 
oblique  line  CE.  With  CU  as  a 
radius  and  C  as  a  center,  draw  an 
arc  cutting  the  line  AB.  Continue 
the  line  OB  till  it  touches  the  arc 
in  a  point  which  we  call  F.  Which  is  longer,  CB  or  OF? 
Compare  OF  and  OF.     Compare  OB  and  OF. 


76  TRIANGLES  AND  LINES, 

81.  Take  some  other  point  in  the  line  AB^  as  the  point 
H^  and  show  that  an  oblique  line  drawn  to  it  from  the  point 
C  is  greater  than  the  i>erpendicular  drawn  from  the  same 
point  to  the  same  line. 

Principle  21.  —  //  from  a  point  without  a  line  a 
perpendicular  and  oblique  lines  are  drawn  to  that  line, 
tJie  perpendicular  is  slwrter  ilian  any  oblique  line. 

82.  Which  is  the  longest  side  of  a  right  tri- 
angle ?  Give  reasons.  In  the  triangle  right- 
angled  at  B^  what  line  is  the  shortest  distance 
from  the  point  A  to  the  line  of  the  base  CB  ? 

83.  Draw  an  acute-angled  triangle  ABC  and  a  dotted 
line  to  show  the  shortest  distance  from  the  point  A  to 
the  line  BC, 

Note.  —  A  perpendicular  drawn  from  the  vertical  angle  of  a  tri- 
angle to  its  base  or  its  base  prolonged  is  called  the  Altitude  of  the 
triangle. 

84.  Reproduce  the  triangle  in  Ex.  83  and  draw  tJie 
altitude  from  the  point  C.     From  the  point  B, 

85.  Draw  a  dotted  line  to  show  the  alti- 
tude from  the  point  B  to  the  base  AC  pro- 
longed ;  the  altitude  from  the  point  C  to  the 
base  AB ;  the  altitude  from  tiie  point  A  to 
the  base  BC  prolonged. 

86.  Draw  a  square  8  inches  each  way  and  its  diagonal. 
What  kind  of  triangles  are  formed  ?  Show  the  base  and 
altitude  of  each.  What  part  of  the  square  is  each  ?  What 
is  the  area  of  each  ? 


TRIANGLES  AND  LINES.  77 

87.  Draw  a  rectangle  8  inches  by  6  inches  and  its 
diagonal.  What  kind  of  triangles  are  formed,  and  what 
is  the  area  of  each  ? 

88.  Is  it  true  that  the  area  of  a  right  triangle  is  equal 
to  one  half  that  of  the  rectangle  having  the  same  base  and 
altitude,  or  that  it  equals  one  half  the  product  of  its  base 
by  its  altitude  ?     Illustrate  your  answer. 

89.  What  is  the  area  of  a  right  triangle  whose  base 
is  17  feet  and  altitude  9  feet  ? 

90.  Find  the  area  of  a  right  triangle  having  a  base  of 
3|-  feet  and  an  altitude  of  16  inches. 

91.  Find  the  area  of  a  right  triangle  having  a  base  of 
7^  feet  and  an  altitude  of  3^  feet. 

92.  Reproduce  the  rectangle  ABB  (7,  making 
AB  6  inches  in  length  and  AO  ^  inches.  Join 
E,  the  middle  point  of  J.5,  with  C  and  D. 
What  kind  of  a  triangle  is  QEB  ? 

93.  Draw  the  dotted  line  EF  perpendicular  to  OB. 
What  is  the  altitude  of  the  triangle  CEB  ?  How  long 
is  EF"^  What  part  of  the  rectangle  EBBF  is  the  tri- 
angle FEB  ?  The  triangle  CEF  is  what  part  of  the 
rectangle  AEFC?  The  triangle  CEB  is  what  part  of 
the  rectangle  ABBC?  What  is  the  area  of  the  triangle 
CEB? 

Note.  —  Dotted  lines  added  to  a  figure  to  help  in  studying  it  are 
called  Construction  Lines. 

94.  Draw  an  isosceles  triangle  and  a  construction  line 
showing  its  altitude.     Draw  construction   lines   forming 


78  TRIANGLES  AND  LINES. 

about  the  triangle  a  rectangle  having  the  same  base  and 
altitude.  If  the  base  is  7  inches  and  the  altitude  10 
inches,  what  is  the  area  of  the  triangle  ? 

95.    Reproduce  the  scalene  triangle  ^  ^ 

ABC^  drawing  the  construction  line  |  ^/^N       • 

BD  to  represent   its   altitude.      Con-  i        y^      \  \ 

struct  the  rectangle  AEFC  about  the  \^y^ \      \ 

triangle.  ^  ^       ^ 

The  triangle  DBC  is  what  part  of  rectangle  BBCFl 

The  triangle  ABB  is  what  part  of  rectangle  EBB  A  ? 

The  triangle  ABC'm  what  part  of  rectangle  AEF01 

Note.  —  You  will  observe  that  we  find  the  areas  of  all  these  differ- 
ent kinds  of  triangles  in  the  same  way,  by  taking  half  the  product 
of  the  base  and  altitude. 

^    ^        Principle  22.  —  The  area  of  a  triangle  is  equal  to 
one  half  the  product  of  its  base  and  altitude. 

96.  Draw  a  triangle  and  show  in  what  way  you  find 
its  area. 

97.  Find  the  area  of  a  triangle  whose  base  is  15  feet 
and  altitude  10  feet. 

98.  The  area  of  a  triangle  is  30  square  feet ;  its  bafie, 
6  feet ;  find  its  altitude. 

99.  The  area  of  a  triangle  is  40  square  feet ;  its  base, 
10  feet ;  find  the  altitude. 

100.  The  area  of  a  triangle  is  48  square  feet ;  the  alti- 
tude, 12  feet ;  find  the  base. 

101.  The  area  of  a  triangle  is  56  square  feet ;  the 
altitude,  8  feet ;  find  the  base. 


TRIANGLES  AND  LINES. 


79 


102.  Draw  a  rectangle  ABCD  8  inches  by  6  inches. 
Join  the  middle  point  of  the  upper  base  with 
the  extremities  of  the  lower  base  i>(7,  mak- 
ing the  triangle  BEO.  Draw  a  diagonal, 
making  the  triangle  BBC.  Join  jP,  a  point 
on  the  upper  base,  with  the  extremities  of 
the  lower  base,  making  the  triangle  BFO. 
What  kind  of  a  triangle  is  each  and  which  is  the  greatest  ? 
Give  reasons. 


103.  (7i>,  4  inches,  is  perpen- 
dicular to  AB^  14  inches,  at  its 
middle  point.  Find  the  area  of 
each  triangle. 


104.  (7  is  the  middle  point  of  J.i>, 
22  inches,  and  of  BE^  12  inches.     The 
angles  at  O  are  right  angles.     Find  the    ^ 
area  of  each  triangle. 

105.  AB^  10  inches,  is  met  perpen- 
dicularly at  its  middle  point  by  (7i>,  5 
inches.  Find  the  area  of  each  of  the 
triangles  thus  formed.  What  kind  of 
triangles  are  they  ? 


106.    What  is  the  altitude  of  a  triangle  ?    Does  the  line 
representing  the  altitude  always  fall  inside  the  triangle  ? 


107.  Draw  a  line  which  represents 
the  altitude  of  the  triangle  J.^  (7  when 
AB  is  considered  the  base  ;  when  AO 
is  considered  the  base  ;  when  BO  is 
considered  the  base. 


80 


TRIANGLES  AND  LINES. 


108.  BD^  a  perpendicular  to  the  line 
AC,  striking  its  prolongation,  is  4 
inches  ;  vl(7  is  5  inches  ;  find  the  area 
of  ABC. 


109.  BC=S  inches;  AD  perpen- 
dicular to  BC  prolonged  =  9  inches  ; 
find  the  area  of  the  triangle. 


110.  AB=  12  inches;  CD  i)erpen- 
dicular  =  4  inches  ;  find  the  area  of 
the  triangle. 

I- 

111.  The  distance  BD  from  the 
point  B  to  AC  prolonged  =  8  inches  ; 
AC  =  5  inches ;  find  the  area  of  triangle 
ACB, 


112.  AB  and  CD  are  parallel. 
Which  of  the  three  triangles 
CAD,   CUD,   or   CBD,   is    the 

greatest  ? 


113.  Draw  an  isosceles  triangle  ABC  and  two  right 
triangles  having  the  same  base  and  equal  area.  Draw 
two  obtuse-angled  triangles  having  the  same  base  and 
equal  area. 

114.  Draw  the  line  AB  8  inches.  With  the  point  A  as 
a  center  with  a  radius  of  6  inches  describe  an  arc.  With 
the  point  -B  iis  a  center  with  a  radius  of  7  inclies  draw  an 


TRIANGLES  AND  LINES.  81 

arc  intersecting  the  first.  Join  (7,  the  point  of  intersec- 
tion, with  A  and  B,  making  the  scalene  triangle  ABC. 
Construct  another  triangle  with  the  base  7  inches,  and  the 
other  sides  8  inches  and  6  inches.  Construct  a  third 
triangle  with  the  base  6  inches,  and  the  other  sides  8 
inches  and  7  inches.  Cut  out  the  three  triangles  and 
superpose  one  upon  another,  and  show  that  they  may  be 
made  to  coincide  in  all  their  parts. 

Principle  23. — Two  triangles  are  equal  when  the 
three  sides  of  the  one  are  respectively  equal  to  the  three 
sides  of  the  other. 

115.  AB  and  BB  are  each  7  inches, 
BE  and  BO  each  5  inches,  AE,  EC,  and 
CD  each  3  inches.     Compare  the  perim- 
eters and  areas  of    the    triangles  ABE  '^' 
and  CBB.  .  Of  ^5(7  and  EBB.  "'       "^ 

116.  With  any  point  as  a  center  and  any  radius, 
describe  an  arc.  Choosing  a  point  in  the  arc,  mark  with 
the  dividers  two  other  points  in  the  arc  equally  distant 
from  the  chosen  point,  and  connect  them  with  it.  Con- 
nect the  three  points  with  the  center  of  the  circle,  and 
show  that  the  triangles  thus  formed  are  equal. 

117.  Join  the  vertex  of  an  isosceles  triangle  with  the 
middle  point  of  the  base,  and  shoAV  that  the  triangles  thus 
formed  are  equal  in  all  their  parts. 

118.  With  a  given  point  as  a  center  and  any  radius, 
draw  an  arc.  Connect  any  two  points  in  the  arc  by  a 
chord,  and  draw  radii  to  its  extremities.  What  kind  of 
a  triangle  is  thus  formed  ?  Join  the  middle  point  of  the 
chord  with  the  center.  Show  that  the  two  triangles  thus 
formed  are  equal  in  all  their  parts. 

HORN.    GEOM. C 


82 


TRIANGLES  AND  LINES. 


119.  Construct  the  angle  ABC  50°,  making  BA  5 
inches  and  BC  7  inches.  Join  AC.  Construct  the  equal 
angle  DEF  with  lines  respectively  equal.  Join  BF. 
Sui)erpo8e  angle  BEF  upon  angle  ABC  so  that  the  5-inch 
lines  shall  coincide  and  the  7-inch  lines  shall  coincide. 
Since  point  B  is  on  point  A,  and  i>oint  F  is  on  point  (7, 
must  not  the  lines  BF  and  AC  coincide?  Are  the  tri- 
angles ecpial  in  all  tlieir  (uirts?  Measure  the  angles,  and 
show  that  the  corresponding  angles  are  equal. 

Note.  —  The  corresponding  angles  of  a  figure  are  called  Homol- 
ogous Angles,  and  the  corresponding  sides  Homologous  Sides. 

Pkincii'LK  24.  — If  two  triangles  have  two  sules  and 
the  included  angle  of  one  respectively  equal  to  two 
sides  and  tlie  included  angle  of  the  otJver,  the  triangles 
are  equfd  in  all  tlieir  parts. 

120.  AB  and  BE  are  each  8  centi- 
meters ;  A  C  and  BF  are  each  7  centi- 
meters. The  angles  A  and  B  are  equal. 
The  perimeter  of  the  triangle  ABC  is  27 
centimeters.     Find  the  side  EF. 

121.  AB  and  BF  are  each 
8  inches  in  length  ;  AC  and 
BE  each  10  inches  ;  EF  is 
6  inches.  The  angles  A  and 
B  are  each  30°.  How  long 
is  the  perimeter  of  ABC? 

122.  Angle  A  =  angle  B. 
AB=^BE;AC=BF;BE 


=   EF-\-4: 


inclies 


BF  = 


BE  -\-l  inches  ;  BE  +  EF 

+  BF  =  ^2  inches  ;  required  BC, 


TRIANGLES  AND  LINES. 


83 


123.  The  lines  AB  and  CD  are  each  12 
inches  long,  and  bisect  each  other  at  E,  The 
angle  BEB  is  60°;  the  side  J.  (7  is  6  inches. 
How  long  is  i>i5?  By  what  geometric  prin- 
ciples can  you  prove  it 

124.  Draw  a  circle  and  two  diam- 
eters AB  and  (7i>,  crossing  at  an  angle 
of  90°.  Join  AC  and  BB.  Can  you 
prove  that  the  triangles  thus  formed 
are  equal  ? 

125.  AB  =  12   inches  ;    CB  =  S   inches ;   ^ 
E  is  the  middle  point  of  each  line,     li  AC 
is  7  inches,  what  is  the  perimeter  of  BEB  ? 
Quote  principle.  ^ 

126.  Draw  the  horizontal  line  AB  6  inches,  and  at  its 
middle  point  C  erect  a  perpendicular  CB  4  inches.  Join 
AB  and  BB.  What  line  is  common  to  both  triangles  ? 
If  AB  is  5  inches,  what  is  the  sum  of  the  perimeters  of 
the  triangles  ? 

127.  Draw  the  vertical  line  AB  10  millimeters  and 
mark  its  middle  point  C.  Draw  the  perpendicular  line 
CB  extending  12  millimeters  to  the  right.  Join  AB  and 
BB.  BB  being  13  millimeters,  what  is  the  sum  of  the 
perimeters  of  the  triangles  thus  formed  ? 

128.  Draw  a  circle  with  a  radius  of  8  centimeters,  and 
at  its  center  make  two  angles  of  120°  each.  Draw  the 
chords  of  the  arcs  which  measure  these  angles  and  show 
that  the  two  triangles  thus  formed  are  equal. 

129.  Draw  the  chord  of  the  remaining  central  angle 
and  compare  the  triangle  thus  formed  with  either  of  the 


84  TRIANGLES  AND  LINES. 

others.     What  kind  of  a  triangle  is  formed  by  the  three 
chords  ? 

130.  Show  how  you  inscribe  an  equilateral  triangle  in 
a  circle. 

Note.  —  A  polygon  is  said  to  be  inscribed  in  a  circle  when  the 
vertices  of  all  the  angles  of  the  polygon  are  in  the  circumference. 

131.  Include  an  angle  of  40°  by  two  lines,  each  6  inches 
long,  and  join  their  extremities.  What  kind  of  a  triangle 
is  thus  formed?  Divide  the  included  angle  into  two 
angles  of  20®  each,  and  continue  the  construction  line 
to  the  opposite  side  or  ba«e.  Show  that  the  two  triangles 
thus  formed  are  equal  in  all  their  parts. 

182.  In  the  isosceles  triangle  ABC  the 
angle  B  is  bisected  by  BD,  What  two 
sides  and  included  angle  of  the  right-hand 
triangle  equal  two  sides  and  the  included 
angle  of  the  other  triangle?  Quote  the 
principle  that  declares  the  equality  of  these 
triangles.  If  the  angle  A  is  75°,  how  many 
degrees  is  the  angle  C  ?  Since  the  two  angles 
at  D  are  equal,  how  many  degrees  are  there 
in  each?  What  kind  of  a  line  is  BDf  If 
AC  \^  10  inches,  how  long  is  -4Z) ? 

Show  the  truth  of  the  following  principle: 

Principle  25.  —  In  an  isosceles  triangle  tJie  line 
which  bisects  the  vertical  angle  bisects  the  base  and  tlie 
triangle,  and  is  perpendicular  to  the  base. 

133.  The  equal  sides  of  an  isosceles  triangle  whose 
perimeter  is  70  inches  are  each  5  inches  longer  than 
the  base.     A  perpendicular  is  dra^vn  from  the  vertex  of 


TRIANGLES  AND  LINES.  85 

the  triangle  to  the  base.      Find   the   distance   from  the 
foot  of  the  perpendicular  to  an  extremity  of  the  base. 

134.  In  an  isosceles  triangle  whose  perimeter  is  bb  inches 
the  base  lacks  5  inches  of  being  as  long  as  one  of  the  equal 
sides.  The  triangle  is  divided  by  a  perpendicular  from 
the  vertex  to  the  base.  Find  the  distance  from  the  ver- 
tex of  a  base  angle  to  the  foot  of  the  perpendicular. 

135.  In  the  isosceles  triangle  whose  perimeter  is  62 
inches  and  whose  equal  sides  are  each  17  inches,  what  is 
the  distance  from  the  vertex  of  the  base  angle  to  the  foot 
of  a  perpendicular  from  the  vertical  angle  to  the  base  ? 

136.  In  the  equilateral  triangle  whose  perimeter  is 
39  inches,  how  long  is  each  segment  of  a  side  cut  off 
by  a  perpendicular  to  that  side  drawn  from  the  vertex 
of  the  opposite  angle  ? 

Note.  —  The  word  "segment"  means  "a  piece  cutoff."  A  seg- 
ment of  a  circle  means  a  piece  cut  off  by  one  line.  One,  straight 
stroke  through  a  sphere  will  cut  off  a  segment  of  a  sphere. 

137.  In  the  isosceles  triangle  ABC  whose  perimeter, 
which  is  60  inches,  is  37  inches  longer  than  one  of  the 
equal  sides,  J.  (7  is  the  base  and  BD  its  bisecting  perpen- 
dicular.    Find  J.i)  and  i>(7. 

138.  In  the  isosceles  triangle  ABC  of  which  AC  i^ 
the  base,  BC  is  13  inches  and  the  perpendicular  BJD, 
drawn  from  the  vertex  to  the  base,  is  12  inches.  The 
perimeter  of  each  of  the  triangles  into  which  it  is  bisected 
by  the  line  BI)  is  30  inches.     Find  the  length  of  the  base. 

139.  The  sum  of  the  equal  sides  of  an  isosceles  triangle 
whose  perimeter  is  40  inches  is  3  times  the  base.  Find 
the  distance  from  the  foot  of  the  perpendicular  which 
bisects  the  base  to  the  vertex  of  the  base  angle. 


86 


TRIANGLES  AND  LINES. 


140.  In  the  triangle  ABCiYiQ  lines  BA 
and  BC  each  meet  the  base  5  inches 
from  the  foot  of  the  perpendicular  BD, 
BA  is  10  inches  long;  how  long  is  BC'i 
Quote  the  geometric  principle. 

Principle  26.  —  //  from  a  point  xvithout  a  line  a 
perpendicular  and  oblique  lines  are  drawn,  two  oblique 
lines  which  nveet  tlie  given  line  at  equal  distances 
front  tlie  foot  of  the  perpendicular  are  equal. 

141.  AD  =  DC;  AC  =  U  inches; 
BD  is  perpendicular  to  AC ;  AB  =  10 
inches ;  required  the  perimeter  of  tri- 
angle ABC. 

142.  BD^  6  inches,  is  perpendicular 
to  ACylQ  inches,  at  its  middle  point; 
AB  is  10  inches;  find  the  sum  of 
perimeters  of  triangles  J^DCand  BDA.  ^ 

143.  2>  is  the  middle  point  iA  AC \ 
BD  is  peri)endicular  to  AC.  If  BC  is  12 
inches  and  DC  \&  (S  inches,  is  the  triangle 
ABC  scalene,  isosceles,  or  equilateral  ? 
Prove. 

144.  Draw  an  indefinite  line,  and  at  the  point  A  in 
it  erect  a  perpendicular  AB  8  inches.  Lay  off  -4C7  6 
inches  on  the  indefinite  line,  and  on  the  other  side  of  A 
lay  off  AD  6  inches.     Compare  DB  and  BC. 

145.  BC  perpendicular  to  AD  at  its 
middle  point  is  |^  of  CD  and  ^  of  BD. 
AC  is  ^  inches.  What  is  the  perimeter 
of  the  triangle  CBD?  Of  the  triangle 
ABC?     Of  the  triangle  ABD?  ^' 


TRIANGLES  AND  LINES, 


8T 


146.  Can  you  show  that  the  points  2>, 
E^  and  F  in  the  line  BB^  drawn  perpen- 
dicular to  A  (7  at  its  middle  point,  are  each 
the  same  distance  from  A  as  from  (7? 
That  AE  =  EQ  and  AF  =  FO  ? 

147.  Can  you  show  that  the  points  D,    a 
E,  and  F  in  the  line  BB,  perpendicular 
to  ^C^  at  its  middle  point,  are  each  the 
same  distance  from  A  as  from  0? 

148.  Can  you  find  a  point  in  the  line  BB  that  is  not 
just  as  far  from  A  as  it  is  from  C  ? 

Principle  27.  — If  a  perpendicular  is  drawn-  at  the 
middle  point  of  a  straight  line,  every  point  in  the 
perpendicular  is  at  the  same  distance  from  one  extrem- 
ity of  that  line  as  from  the  other. 

149.  Draw  the  line  AB.  With  the  point  J.  as  a 
center,  with  a  radius  greater  than  half  of  AB  describe  an 
arc.  With  the  point  ^  as  a  center,  with  the  same  radius 
describe  an  arc  intersecting  the  first  arc  above  the  line 
AB  and  also  below  it.  Join  the  points  of  intersection. 
Can  you  see  why  the  line  which  joins  the  intersections 
bisects  the  line  AB  ? 

150.  Find  the  area  of  an  isosceles  triangle  whose  perim- 
eter is  64  inches,  one  of  whose  equal  sides  is  25  inches, 
and  whose  altitude  is  24  inches. 

151.  Find  the  approximate  area  of  an  equilateral  tri- 
angle whose  perirbeter  is  120  inches,  and  whose  altitude 
is  34.6+  inches. 

Query.  —  Why  can  we  only  find  the  approximate  area  of  such  a 
tnangle  ? 


88 


TRIANGLES  AND  LINES. 


152.  The  perimeter  of  the  sciilene 
triangle,  right-angled  at  B^  is  36  inches, 
BC  is  one  foot  in  length  and  ^^  is  |  of 
a  foot.     Find  the  area  of  the  triangle. 

153.  AB  is  perpendicular  to  BC; 
angle  BAB  =  50® ;  BD  is  perpendicu- 
lar to  ^C. 

How  many  degrees  has  each  angle  of 
the  triangle  ABI)  ?  Of  triangle  BBC? 
Of  triangle  ABC  f  What  angle  is  com- 
mon to  triangle  ABB  and  triangle  ABC? 

154.  AB  and  CD  are  perpendicu- 
lar to  BE, 

Angle  m  =  55®.  Find  the  other  an- 
gles of  the  triangles  ABE  and  CBE. 
What  angle  is  common  to  both  tri- 
angles ? 

155.  AB  is  perpendicular  to  BC; 
BE  is  parallel  to  AC. 

Angle  I  =  35".  Find  the  other  angles 
of  the  triangles  ABC  and  BBE,  What 
angle  is  common  to  both  triangles  ? 

156.  AB  is  perpendicular  to  BB ;  AB 
=  6  inches ;  BB  =  S  inches ;  BC  =  CB. 
Find  the  area  of  triangles  ABC  and 
ACB. 


CHAPTER   VII. 

CUMULATIVE  REVIEW  NO.   2. 

1.  How  many  boundaries  has  a  parallelopiped  ? 
Note.  —  The  boundaries  of  a  solid  are  called  Surfaces. 

2.  What  forms  the  boundaries  of  a  surface  ? 

3.  What  are  the  extremities  of  lines  ? 

4.  A  point  is  that  which  has  position  without  length, 
breadth,  or  thickness.     Did  you  ever  see  a  point  ? 

5.  A  line  has  length  without  breadth  or  thickness. 
How  do  we  represent  lines  ? 

6.  A  surface  has  length  and  breadth  without  thick- 
ness. How  many  surfaces  has  a  cube  ?  How  many  has 
a  sphere  ? 

7.  What  name  is  given  to  those  geometric  forms  that 
have  length,  breadth,  and  thickness  ? 

Note.  —  A  geometric  form  having  length,  breadth,  and  thickness 
is  called  a  Solid. 

Query.  —  Is  the  word  "solid"  used  here  in  its  ordinary  sense? 

8.  If  a  point  were  moved  in  one  direction  through 
space,  what  would  its  path  be  ? 

9.  What  would  be  the  path  of  a  line  which  was  moved 
through  space  in  the  direction  of  its  length  ?  What 
would  be  the  path  of  a  line  moved  in  the  direction  per- 
pendicular to  its  length? 

89 


90  CUMULATl/E  REVIEW. 

10.  Move  a  piece  of  paper  through  space  in  such  a  way 
that  the  path  of  one  of  its  surfaces  would  be  a  geometric 
solid. 

11.  Cut  out  a  square  and  lay  it  on  the  desk.  If  it  were 
to  rise  in  the  air  keeping  it«elf  always  parallel  to  the 
desk,  what  geometric  solid  would  it  form  when  it  had 
reached  a  height  equal  to  one  of  its  sides  ?  What  would 
you  call  it  when  it  had  reached  a  height  greater  or  less 
than  one  of  its  sides  ? 

12.  If  a  circle  should  rise  in  the  same  way, 
what  geometric  solid  would  be  formed? 

Note.  —  Such  a  geometric  solid  is  called  a  Cylinder. 

13.  Cut  out  a  rectangle  whose  longer  sides  are 
each  22  inches  and  shorter  sides  each  8  inches. 
Place   the  shorter  sides  togetlier  in  such  a  way 

that  the  longer  sides  form  parallel  circumferences.  If 
circles  were  fitted  to  tliese  circumferences,  what  kind  of 
a  solid  would  be  formed  ?  What  would  be  the  area  of  the 
curved  surface  ?  What  is  the  length  of  the  longest  line 
that  can  be  drawn  on  either  of  its  plane  surfaces  ? 

14.  In  what  geometric  form  are  measures  of  bushels, 
half  bushels  and  pecks  generally  made  ? 

16.  If  the  diameter  of  the  bottom  of  a  cylindrical  tin 
pail  is  10  inches,  what  is  its  circumference?  If  ita  height 
is  10  inches,  how  many  square  inches  of  tin  were  used  in 
making  ita  curved  surface  ? 

16.  A  straight  line  is  one  that  does  not  change  its 
direction  at  any  point.     What  is  a  curved  line  ? 

17.  Classify  lines  and  define  each  class. 

18.  Classify  angles  and  define  each  class. 


CUMULATIVE  REVIEW,  91 

19.  What  is  the  complement  of  an  angle  of  48°  50'? 

20.  Work  Ex.  19,  substituting  "supplement"  for 
"complement." 

21.  How  many  degrees  are  there  in  angle  x  if  its  adja- 
cent supplementary  angle  is  50°  larger  ? 

22.  How  many  degrees  are  there  in  an  angle  whose 
complement  is  10°  more  than  3  times  as  large  ? 

23.  How  many  degrees  are  there  in  each  of  the  three 
angles  formed  on  the  same  side  of  a  straight  line  at  a 
given  point  of  that  line  if  the  first  is  10°  greater  than  the 
second,  and  the  second  10°  greater  than  the  third  ? 

24.  ^^  =  12  inches;       DB  =  AC;    a> ^ ^b 

^6^=  4  inches;  Find  CD. 

25.  CD  =  6  inches  ;  AO=DB; 

AB  =  10  inches ;  Find  A  0  and  DB. 

26.  ^i>  =  12  inches  ;      ~  AC=J)B; 
AC  =S  inches;  Find  CB. 

27.  AI)=  CB;  I)B=S  inches  ; 
CD  =  7  inches  ;  Find  AB  and  AB, 

28.  If  the  area  of  the  rectangle  =  70 
square  inches,  what  is  the  value  oi  x? 


29.    Given,  ^^  =  6  inches  ;  AB=^7      ^ 
inches ;  area   of  rectangle  =  Q6  square 
inches  ;  required  BC. 


30.    Find  the  area  and  perimeter  of  a 
rectangle  each  of  whose  long  sides  is  7  centimeters  more 
than  three  times  as  long  as  a  short  side,  and  the  sum  of 
two  adjacent  sides  of  which  is  27  centimeters. 


92  CUMULATIVE  REVIEW. 

31.  Given  a  rectangle  12  millimeters  long  and  10  milli- 
meters wide;  find  the  area  and  perimeter  of  a  rectangle 
drawn  within  it,  having  it«  sides  parallel  to  the  sides  of 
the  original  rectangle  and  at  a  distance  of  2  millimeters 
from  them. 

Query.  —  How  much  shorter  is  each  side  of  the  small  rectangle 
than  the  homologous  side  of  the  Urge  rectangle  ? 

82.  A  rectangular  garden  48  feet  by  36  feet  has  a 
border  3  feet  wide  within  its  limits.  How  much  surface 
is  inside  the  border?  How  much  is  occupied  by  the 
border  ? 

33.  A  square  garden,  whose  dimensions  are  8  yards, 
has  a  square  flower  bed  in  its  center  whose  area  is  |  that 
of  the  garden,  and  whose  sides  are  parallel  to  those  of  the 
garden.  How  long  is  one  side  of  the  flower  bed,  and  how 
far  is  the  middle  point  of  one  of  its  sides  from  the  middle 
points  of  the  parallel  sides  of  the  garden  ? 

34.  What  is  the  sum  of  two  lines  8  feet  and  5  feet? 
Draw  a  figure  on  the  scale  of  a  centimeter  to  the  foot 
showing  the  sum  of  the  squares  of  those  lines,  and  another 
showing  the  square  of  their  sum. 

35.  Find  the  area  of  the  rectangles  and  small  square 
which  must  be  added  to  a  10-inch  square  to  form  a  15- 
inch  square  ;  to  form  an  18-inch  square. 

36.  Given  a  10-inch  square  to  make  into  a  larger 
square.  Find  by  trial  how  wide  the  additions  must  be 
in  order  that  the  rectangles  and  small  square  shall  contain 
96  square  inches. 

37.  Given  a  10-inch  square  as  a  basis  for  a  larger 
square.  Find  by  trial  how  wide  the  additions  must  be 
that  they  may  contain  69  square  inches. 


20  in. 

X 

CUMULATIVE  REVIEW.    '  93 

38.  Given  a  20 -inch  square  to  build  a  larger  one.  Find 
by  trial  the  width  of  the  necessary  additions  if  they  con- 
tain 176  square  inches. 

39.  Find  the  length  of  x  when  the  sum  of 
the  three  additions  to  the  20-inch  square  is 
384  square  inches. 

40.  If  X  represents  the  side  of  a  square, 
what  represents  the  area  of  the  square  ? 

41.  If  a^  represents  the  area  of  a  square,  what  repre- 
sents a  side  ? 

42.  Find  the  area  of  a  square  whose  perimeter  is  100 
inches. 

43.  Find  the  perimeter  of  a  square  whose  area  is  64 
square  inches. 

44.  Divide  a  semicircle  into  two  sectors,  one  of  which 
is  5  times  the  other.  How  many  degrees  are  there  in  the 
angle  of  each  ? 

45.  The  diameter  of  a  circle  is  42  inches.  Find  the 
length  of  the  arcs  of  two  sectors  which  compose  the  circle, 
the  greater  being  10  times  the  less.     Illustrate. 

46.  An  arc  which  is  a  quadrant  is  16|  feet.  Find  the 
diameter. 

47.  What  angle  is  the  greatest  common  measure  of  a 
given  angle  and  its  complement,  if  the  complement  con- 
tains 24°  more  than  the  given  angle  ? 

Suggestion.  —  Let  x  =  given  angle. 

48.  What  angle  is  the  G.  C.  M.  of  a  given  angle  and 
its  supplement,  if  the  supplement  is  30°  more  than  twice 
the  given  angle  ? 


94 


•     CUMULATIVE  REVIEW. 


49.  What  angle  is  the  G.  C.  M.  of  angle 
a  and  angle  6,  made  by  a  transversal  to  two 
parallels,  if  angle  b  lacks  95°  of  being  4 
times  angle  a?  How  many  times  is  the 
measure  contained  in  angle  c?     In  angle  d? 

50.  Define  each  t«rm  in  the  following  classification  of 
triangles :     Equilateral.     Isosceles.     Scalene. 

51.  Classify  triangles  with  regard  to  their  angles. 

52.  The  shortest  side  of  a  scalene  triangle  is  7  inches 
less  than  the  one  of  medium  length,  and  the  one  of 
medium  length  is  7  inches  shorter  than  the  longest.  The 
perimeter  is  93  inches.     Find  each  side. 

58.  Draw  a  right  triangle  with  a  base  of  6  centimeters 
and  an  altitude  of  4  centimeters,  and  find  its  area  in 
square  millimeters. 

54.  In  the  right  triangle  ABC,  AB  =  12 
inches,  BC=S  inches,  ^7>  =  6  inches,  BE  =^  A 
inches.  The  triangle  DBE  is  what  fractional 
part  of  the  triangle  ABCl 

SutiUK8TioN.  —  Compare  the  haae  and  altitude  of 
the  small  triangles. 

55.  The  vertical  angle  of  an  isosceles  triangle  is  30® 
larger  than  eitlier  of  the  base  angles.     Find  each  angle. 

56.  ^^(7  is  an  isosceles  triangle  ;  AB  '^^  *^ 
is  the  base.     The  angle  A  CB  is  15°  more 
than  twice  ABC.     Find  all  the  angles. 

57.  In  the  circle,  whose  center  is  0, 
angle  AOCi&  112°.  How  many  degrees 
is  angle  BOCl  Angle  OCB't  Angle 
OBCi 


CUMULATIVE   REVIEW. 


95 


58.  In  the  circle,  whose  center  is  0, 
angle  AOO  is  70°.  How  many  degrees 
is  angle  ^0(7?  Angle  OCBl  Angle 
OBCl  A 

59.  In  the  circle,  whose  center  is  0, 
angle  AOQ  \^  16°  less  than  twice  angle 
COB.  How  many  degrees  is  angle 
OCB"^ 

60.  In  the  circle,  whose  center  is  0, 
the  angle  AOB  is  110°.  How  many 
degrees  has  each  angle  of  the  triangle  ? 
How  many  has  angle  BOC^ 

61.  Each  of  the  equal  sides  of  an  isosceles  triangle  is 
13  inches,  and  its  perimeter  is  36  inches.  The  perpen- 
dicular from  the  vertex  to  the  base  is  12  inches.  Find 
the  area  and  perimeter  of  the  triangle  formed  on  each  side 
of  the  perpendicular. 


62.  The  quadrilateral  ABD  C  has  the 
sides  AB  and  CD  parallel.  To  what 
special  name  is  it  entitled  on  that  ac-  c 
count  ?  If  .4.  (7  is  4  inches,  BD  1  inch  longer  than  A  C, 
AB  twice  AC^  and  CD  twice  BD^  how  long  is  the  perim- 
eter of  the  trapezoid? 

63.  How  many  degrees  are  there  in  each  of  five  equal 
angles  formed  about  a  given  point  ? 

64.  How  many  degrees  are  there  in  the  angle  formed 
by  the  bisector  of  an  angle  of  50°  and  the  bisector  of  its 
supplement?  • 


96  CUMULATIVE  REVIEW. 

65.    AB  is  parallel  to  CD. 

Angle  KMB  =  68°. 

FE  is  a  bisector  of  angle  KMB, 

GH  is  parallel  to  FE.   Find  angle  HLD. 

Q^.  Include  an  acute  angle  by  two  lines,  one  5  inches, 
the  other  8  inches.  Join  their  extremities.  With  the 
joining  line  as  a  base  draw  arcs,  and  construct  a  triangle 
having  an  8-inch  side  adjacent  to  the  5-inch  side  of  the 
first  triangle.  Erase  the  joining  line,  and  name  the  quad- 
rilateral tlius  formed. 

67.  Draw  a  rhomboid  whose  longer  sides  are  each  3 
times  a  shorter  side  and  whose  i>erimeter  is  8  inches,  and 
write  u}K)n  each  side  its  length. 

68.  Draw  a  rei*tangle  which  is  the  product  of  a  line  12 
centimeters  by  a  line  3  centimeters.  Also  one  which  is 
the  square  of  a  line  6  inches.  Which  has  the  greater 
area?     The  greater  perimeter? 

60.  Which  will  require  more  fencing,  a  rectangular  lot 
8  rods  by  2  rods,  or  a  square  lot  containing  the  same  area  ? 

A    »tn. 

70.  The  angles  being  all  right  angles,  how 
long  is  the  line  AHf  Find  the  area  and 
perimeter  of  the  surface  inclosed.  jj 


lin. 

iL 


tin. 


CHAPTER   VIII. 
QUADRILATERALS. 


Trapezium.  Trapezoid.  Kuomboiu.  Kiiombus. 

1.  Draw  a  quadrilateral  having  no  two  sides  parallel 
and  write  its  name  upon  it. 

Note.  —  A  quadrilateral  having  no  two  sides  parallel  is  called  a 
Trapezium. 

2.  Find  the  perimeter  of  the  trapezium  ABCD,  AB 
being  twice  BC^  which  is  10  inches,  AD  being  15  inches 
longer  than  BC^  CD  being  4  times  BQ. 

3.  Find  the  perimeter  of  the  trapezium  ABCD^  whose 
side  AB  is  8  inches,  AB  4  inches  longer  than  AB^  DO  2 
inches  longer  than  AD^  BQ  twice  AB. 

4.  In  the  trapezium  ABCD^  whose  perimeter  is  50 
inches,  the  side  AB  is  twice  BQ,  AD  is  3  times  BQ, 
and  QD  is  4  times  BQ.     How  long  is  each  side  ? 

Query.  —  a;  =  ? 

5.  In  the  trapezium  ABQD, 

AD  =  4:  times  AB. 
QD  =  S  times  AB. 
BQ=:9  inches  more  than  AB, 
The  perimeter  of   the  trapezium  is  63  inches.     Find 
each  side. 

HORN.  GEOM. 7  97 


98 


QUADRILA  TEHALS. 


6.  The  trapezium  ABCD  is  divided  by 
the  diagonal  AC.  What  is  the  sum  of 
tlie  angles  of  the  triangle  ABC?  Of  the 
triangle  ADC?  How  many  degrees  are 
there  in  all  the  angles  of  the  quadrilateral  ? 
Can  you  draw  a  quadrilateral  which  cannot  be  divided 
into  two  triangles  by  a  diagonal  ? 

PiiiNCiPLE  28.  —  TVmj  sum  of  all  the  angles  of  a 
quadrilateral  is  equal  to  four  right  angles  or  to 
360^' 

7.  In  the  trapezium  ABCD  angle  A  is  60%  angle  B  is 
I  of  angle  yl,  angle  C  is  20*  more  than  angle  J5.  Find 
angle  D, 

8.  In  the  trai>czium  ABCD  angle  A  =  2  times  angle 
B\  angle  C=  angle  B -\- W  \  angle  i>  =  angle  ^  +  20^ 
Find  each  angle. 

SUOGK8TION.  — Let  X  —  angle  B. 

9.  One  of  the  exterior  angles  of  a  trapezium  is  79% 
another  exterior  angle  is  93%  another  84**.  Find  all  the 
angles  of  the  tra|>ezium. 

10.  Find  the  area-  of  the  trapezium 
ABCD^  if  the  diagonal  AC  \^  1  inches, 
the  altitude  BE  4  inches,  the  altitude  CH 
6  inches,  and  the  side  AD  10  inches. 

11.  AB  =  12  centimeters. 
The  diagonal  AD  =  1  decimeter. 
The  perpendicular  from  D  to  AB  is  6 

centimeters.      The  perpendicular  from    C 

to  ^Z>  =  8  centimeters.     Find  the  area  of  the  trapezium. 


QUADRILATERALS.  99 

12.  Draw  a  trapezium,  divide  it  into  triangles,  meas- 
ure their  bases  and  altitudes,  and  find  the  area  of  the 
trapezium. 

13.  Draw  a  trapezium  and  a  trapezoid  and  tell  what 
distinguishes  one  from  the  other. 

14.  Draw  a  diagonal  to  a  trapezoid  and  show  how 
many  degrees  there  are  in  the  sum  of  all  its  angles. 

15.  AB  and  OD  are  each  perpen-      ^ ^ 

dicular  to  AQ.  Angle  ^  is  3  times 
angle  D.  How  many  degrees  are 
there  in  each  angle  of  the  trapezoid  ? 

16.  Find  the  perimeter  of  a  trapezoid  one  of  whose 
parallel  bases  is  1^  times  the  length  of  the  other,  which  is 
6  inches,  the  sum  of  the  non-parallel  sides  being  21 
inches. 

Note.  —  Parallel  sides  of  a  trapezoid  are  called  Bases. 

17.  In  the  trapezoid  ABCD  the  sum  of  the  bases  is 
18  inches  and  the  greater  is  twice  the  less  ;  the  sum  of 
the  non-parallel  sides  is  9  inches  and  the  greater  is  twice 
the  less.  Find  the  perimeter  and  each  side  of  the 
trapezoid. 

18.  Draw  an  isosceles  triangle  whose  base  angles  are 
each  80°,  and  divide  it  by  a  line  drawn  parallel  to  the 
base.  What  figures  are  formed?  How  many  degrees  are 
there  in  each  angle  of  the  upper  figure  ?  In  each  angle 
of  the  lower  figure  ? 

Note.  —  When  a  trapezoid  has  its  non-parallel  sides  equal  it  is 
called  an  Isosceles  Trapezoid.  The  non-parallel  sides  are  called 
Legs. 

19.  One  base  of  an  isosceles  trapezoid  is  11  centi- 
meters, the  other  is  7|-  centimeters  more  than  one  half  as 


100  QUADRILATERALS. 

long,  and  each  leg  is  7  centimeters  shorter  than  the  greater 
base.     Find  the  j)erinieter. 

20.  Draw  an  isosceles  trapezoid  and  prolong  the  non- 
parallel  sides  until  they  meet.  What  figure  have  you 
formed  ? 

21.  If  your  sister  is  10  years  old  and  your  brother  is 
14  years  old,  what  is  their  average  age  ? 

22.  If  you  are  marked  90%  on  one  examination  paper 
and  100%  on  another,  wliat  is  your  average  %  ? 

23.  If  a  lioard  is  20  inches  wide  at  one  end  and  10 
inches  at  the  other,  what  is  its  average  width  ?    Illustrate. 

24.  Draw  the  trapezoid  ABCD^  making  AB  8  inches, 
2>(7  10  inches,  distance  between  bases 
5  inclies.  Wliat  is  the  length  of  XY 
drawn  midway  between  tlie  bases? 
Through  Y  draw  NM  perpendicular 
to  the  bases.  Cut  off  the  triangle 
YMC  and  place  it  in  the  position  of  YNB,  What  b  the 
area  of  the  rectangle  thus  formed  ?  How  does  it  compare 
with  the  tra^Hizoid  ? 

Note.  —  A  line  drawn  midway  between  the  bases  of  a  trai>ezuid 
is  called  a  Median. 

25.  AB,    10    inches,   and    CD,   12      a b 

inches,  are  bases  of  a  trapezoid.    Find    xh \r 

the  length  of  the  median,  XY.  J- ^c 

26.  The  distance  between  the  bases  of  a  trapezoid  is 
14  centimeters,  and  the  median  is  14  centimeters.  How 
many  square  centimeters  are  there  m  its  surface  ? 

XoTE.  —  The  iier]>endicular  distance  between  the  bases  of  a  trape- 
zoid is  called  its  Altitude. 


QUADRILATERALS. 


101 


27.  AB  =  6  inches  ;  CD  =  8  inches  ;  alti- 
tude =13  inches.  What  is  the  area  of  the 
trapezoid  ?  What  is  the  perimeter  of  the  rec- 
tangle having  the  same  altitude  and  area  ? 


28.  AB  =  10  inches  ;  JDC  =  12  inches  ;  AU,  the  alti 
tude,   =  4  inches.     Find  the  area  of       ^  ^ 
the  trapezoid.     Reproduce  the  trape- 
zoid, cut  and  replace  the  parts  neces- 
sary to  transform  it  into  a  rectangle. 
How  do  you  find   the  average  length  of   the  bases  of  a 
trapezoid  ? 

Principle  29.  —  The  area  of  a  trapezoid  is  equal  to 
the  product  of  its  altitude  and  one  half  the  sum  of 
its  parallel  sides. 

29.  How  many  square  feet  has  a  plank  20  feet  long, 
1  foot  wide  at  one  end  and  1  foot  6  inches  at  the  othet  ? 

30.  AB  =  18  centimeters; 
C7)  =  12  centimeters;  jE^jP,  the 
altitude,  =  125  centimeters. 
Find  the  area  of  the  trapezoid. 

31.  The  sum  of  the  bases  of  a  trapezoid  is  27  inches ; 
the  altitude  is  5  inches.     Find  the  area. 

32.  The  lower  base  of  a  trapezoid  is  5  inches  more 
than  3  times  the  upper  base,  and  the  sum  of  the  bases 
is  37  inches.  Find  each  base.  The  altitude  is  f  of  the 
upper  base.     Find  the  area. 

33.  The  lower  base  of  a  trapezoid  is  6  inches  less  than 
twice  the  upper  base,  and  their  sum  is  15  inches.  The 
altitude  is  ^  the  lower  base.     Find  the  area.     If  one  of 


102  QUADHILA  TERALS. 

the  non-parallel  sides  is  5  inches  and  the  other  is  f  of 
the  upper  base,  what  is  the  perimeter  of  the  tra})ezoid  ? 

34.    Into  what  triangles  does  the  diagonal  AC  divide 
the   tra|)ezoid  ABCD?     Draw  a  con- 
struction line  showing  the  altitude  of        /^^^I~~^ 
tlio   triangle   ABC  when    BC   is    the      /  ^"^^^ 

\n\He.     Sliow  the  altitude   of   the   tri- 
angle ADC  when  AD  is  the  base      Which  altitude  is  the 
greater  ?     Give  reaAons. 

85.  AD  is  10  inches,  BC  14  incites,  and  the  |)er|)endic- 
ular  distanrt*  lietween  the  bases  «5  inches.  Find  the  area 
of  each  triangle  and  add  them  to  fnid  the  area  of  the 
traiiezoid.  Which  plan  of  finding  the  area  of  a  trape- 
zoid do  you  prefer,  and  why? 

SG.  The  sum  of  the  Uises  of  a  tra|)ezoid  is  15  incites, 
the  upiH»r  base  being  twice  the  lower  ;  the  altitude  is  4 
inrlies.  Find  the  area  of  each  triangle  into  which  a  diago- 
nid  divides  the  tra])ezoid. 

87.  AB  =  8  inches ;  GH  =  14  inches ; 
CD  =  4  inches  ;  EF  ^  1  inches.  The 
altitude  of  the  greater  tra|)ezoid  is  10 
inches,  of  the  less,  5  inches.  Find  the 
ninnlwr  of  square  inches  in  the  s^iace  in- 
cluded between  their  perimeters. 

38.  The  bases  of  a  trajjczoid  are  9  inches  and  13  inches. 
The  area  is  55  square  inches.  How  far  ajMirt  are  the 
bases?  What  is  their  average  length?  How  does  the 
difference  between  the  shorter  bjise  and  their  average 
length  compare  with  the  difference  between  their  average 
length  and  the  longer  base  ? 


QUADRILA  TERALS.  103 

39.  The  area  of  a  trapezoid  =  45  square  inahes  ;  greater 
base  =  12  inches ;  altitude  =  5  inches.   Find  the  smaller  base. 

40.  The  area  of  a  trapezoid  is  52  square  inches ;  the 
average  length  of  the  bases  is  13  inches,  and  the  greater 
base  is  16  inches.     Find  the  other  base  and  the  altitude. 

41.  How  much  surface  is  covered  by  a  bay  window  in 
the  shape  of  a  trapezoid,  the  longer  of  the  parallel  sides 
being  12  feet,  the  shorter  6  feet,  and  the  distance  between 
them  3  feet  ? 

42.  The  irregular  hexagon  ABCDEF 
has  the  sides  AB  8  inches  and  ED  6 
inches,  each  parallel  to  the  diagonal  FC^ 
12  inches.  GH^  perpendicular  to  FC^  is  f 
7  inches ;  UK,  the  prolongation  of  GIT^ 
is  5  inches.     Find  the  area  of  the  hexagon. 

43.  Draw  two  equal  parallel  lines  and  join  each  ex- 
tremity of  one  to  that  extremity  of  the  other  which  is  the 
nearer.  The  surface  thus  inclosed  is  bounded  by  how 
many  lines  ?  Are  both  pairs  of  opposite  sides  parallel  ? 
To  what  name  is  the  figure  entitled  by  the  parallelism 
of  its  opposite  sides  ? 

Suggestion.  —  See  Chap.  V.,  p.  52. 

44.  Find  the  perimeter  of  a  parallelogram,  in  which 
one  side  is  6  inches,  and  the  sides  adjacent  to  it  are  each 
2 J  times  as  long. 

45.  Draw  a  rectangular  parallelogram  whose  adjacent 
sides  are  7  centimeters  and  5  centimeters,  and  find  its 
perimeter  and  area. 

46.  Draw  a  parallelogram,  not  rectangular,  the  longer 
sides  of  which  are  each  twice  a  shorter  side,  tlie  shorter 


-1         G         B 

J 

" 

\ 

/ 

/ 

104  QUADRILA  TERALS. 

sides  being  each  2  centimeters.  How  long  is  the  perimeter 
of  the  parallelogram  ?  What  name  is  given  to  a  parallelo- 
gram whose  angles  are  oblique  ? 

47.  Find  the  perimeter  of  a  rhomboid  each  of  whose 
long  sides  is  equal  to  the  sum  of  the  shorter  sides,  a  short 
side  being  20  millimetei*s. 

48.  The  perimeter  of  a  rhomboid  is  66  feet.  Each 
long  side  is  3  times  a  short  side.     Find  each  side. 

49.  Is  a  rectangle  a  parallelogram  ?  Is  a  square  ? 
Give  reasons. 

50.  AB  and  CD  are  parallel.  What 
name  belongs  to  AC'(  If  angle  x  =  87", 
what  is  angle  y't  Quote  a  geometric 
principle  which  helps  you  to  your  answer. 

51.  AB  and  CD  are  parallel.  ED  is 
parallel  io  AC  If  a:  =50°,  how  many 
degrees  is  y'f  How  many  is  2?  Com- 
pare angle  x  and  angle  z. 

52.  In  the  parallelogram  ABCDy  if 
ar=  60°,  how  many  degrees  is  the  angle 
y  ?     The  angle  z  ?  ^ % 

63.  Draw  a  parallelogram  and  show  the  truth  of  the 
following  principle ; 

PiiiNrrPLE  30. — Tlie  opposite  angles  of  a  parallelo^ 
grain  are  equal. 

54.    The  opposite  sides  of  a  parallelo-    '*t^:^- ^ 

gram    are   equal.     Are    the   triangles       \^^^'"""'"^^\ 

ACD   and   ABD   equal    in    all    their         ^ — ^z> 

parts  ?  Show  tlie  truth  of  your  answer  by  referring  to  a 
geometric  principle. 


B    K 

/. 

f 

A 
B 

c 

D 

/: 

J 

4 

» 

a 

o/" 

0' 

f 

i 

4 

/ 

;/ 

,/ 

^i 

QUADRILATERALS.  105 

55.  AB  and  CD  being  parallel,  what 
is  the  sum  of  the  angles  a  and  h  ? 
Quote  the  geometric  principle. 

56.  AB  and  CD  being  parallel,  how 
many  degrees  are  there  in  angle  x  + 
angle  y  ? 

57.  How  many  degrees  are  there 
in  all  the  angles  of  the  parallelogram 
ABCB}     Give  reasons. 

58.  One  of  the  angles  of  a  rhomboid  is  73°.  Find  the 
number  of  degrees  in  each  of  the  remaining  angles. 

59.  One  of  the  angles  of  a  rhomboid  is  tAvice  an  adja- 
cent angle.  How  many  degrees  are  there  in  each  angle 
of  the  rhomboid? 

60.  One  of  the  angles  of  a  parallelogram  is  3  times  an 
adjacent  angle.  How  many  degrees  are  there  in  each 
angle  of  the  parallelogram  ? 

61.  Can  you  show  that  the  sum  of  all  the  angles  of  a 
parallelogram  equals  360°,  using  the  figure  in  Ex.  57  ? 

62.  One  of  the  angles  of  a  parallelogram  is  equal  to  its 
adjacent  angle.  How  many  degrees  has  each  angle  of  the 
parallelogram  ?     What  kind  of  a  parallelogram  is  it  ? 

63.  Draw  a  rhomboid,  having  its  longer  sides  each  10 
inches  and  its  shorter  sides  each  25%  of  the  length  of  a 
longer  side,  and  find  its  perimeter. 

64.  Make  an  angle  of  80°  by  two  lines,  one  5  inches 
and  the  other  4  inches.  Draw  parallels  to  those  lines, 
forming  a  rhomboid.  How  many  degrees  has  each  of  the 
other  angles  of  the  rhomboid  ? 


106  QUADRTLA  TERMS, 

65.  The  sum  of  the  obtuse  angles  of  a  rhomboid  is 
twice  the  sum  of  the  acute  angles.     Find  each  angle. 

66.  How  does  a  rhomboid  differ  from  a  rectangle  ? 

67.  If  angle  a,  formed  by  the  diagonal    ^^ --^ 

and  one  side  of  the  rhomboid,  is  30%  how    \  .^^x^  \ 

many  degrees  is  angle  6  ?  If  angle  c  is  100°,       i~^ 

how  many  is  angle  dl    Give  authority.     If  angle  x  is  50®, 
how  many  is  angle  y  ?     Quote  geometric  principle. 

68.  The  longer  sides  of  a  rhomlmid  are  each  8  centi- 
meters longer  than  a  short  side,  and  the  i>erimeter  is  36 
centimeters.     Find  the  sides. 

69.  The  longer  sides  of  a  rhomboid  are  each  3  inches 
more  than  5  times  {is  long  as  a  shorter  side,  and  the  ]>erim- 
eter  is  54  inches.     Find  all  the  sides. 

70.  Reproduce  ABCE,  draw  the 
I)erpendiciilar  BD,  cut  off  the  tri- 
angle BDCy  and  place  it  in  the  posi- 
tion of  ALT.  What  is  the  area  of  the  rectangle  ABBF? 
Compare  it  with  the  parallelogram  ABCE.  Comi>are  the 
base  and  altitude  of  the  rectangle  with  the  base  and  alti- 
tude of  the  parallelogram. 

NoTK.  —  The  distance  between  parallel  sides  of  a  parallelogram  is 
called  the  Altitude. 

71.  Can  a  parallelogram  have  more  than  one  altitude  ? 
Illustrate. 

Principle  31. —  The  area  of  a  parallelogram  is  equal 
to  thai  of  a  rectangle  Jiavlng  the  same  hose  and  altitude. 

72.  AB  =  11  inches.  FF,  which 
measures  the  distance  between  AB 
and  C2>,  is  5  inches.  Find  the  area 
of  the  rhomboid. 


le    4 iiiS: ^ 

i-  rV-^u^. ir^' 


oL — 1 — i 


QUADRILA  TEUALS.  107 

73.  Find  the  area  of  a  rhomboid  whose  perimeter  is  36 
millimeters,  whose  shorter  sides  are  each  6  millimeters, 
and  whose  altitude  perpendicular  to  a  long  side  is  4 
millimeters. 

74.  The  perimeter  of  a  rhomboid  =  42  inches  ;  a  long 
side  is  twice  an  adjacent  side  ;  the  altitude  perpendicular 
to  a  long  side  =  6  inches.     Find  the  area  of  the  rhomboid. 

75.  The  area  of  a  parallelogram  =  54  centimeters  ;  the 
base  =  9  centimeters.     Find  the  altitude. 

76.  A  base  of  a  rhomboid  is  4  centimeters  longer  than 
the  altitude  perpendicular  to  it,  and  their  sum  is  16  centi- 
meters.    Find  the  area  of  the  rhomboid. 

77.  A  base  of  a  rhomboid  plus  the  altitude  perpen- 
dicular to  it  equals  32  inches.  The  base  is  8  inches  longer 
than  the  altitude.     Find  the  area  of  the  rhomboid. 

Suggestion.  —  Let  x  =  altitude ;  ?  =  base. 

78.  A  base  of  a  rhomboid  is  5  inches  longer  than  the 
altitude  perpendicular  to  it.  The  same  base  plus  the  same 
altitude  is  19  inches.     Find  its  area. 

79.  The  long  diagonal  of  a  rhomboid  is  12  inches. 
One  of  the  short  sides  is  one  half  the  diagonal,  and  the 
perimeter  of  the  triangle  formed  by  the  diagonal  and  two 
adjacent  sides  is  29  inches.  Find  the  perimeter  of  the 
rhomboid. 

80.  AB  —  5  inches,  and  EH^  perpendicular 
to  it,  =  8  inches.  What  is  the  area  of  the 
rhomboid  ABCD?  What  is  the  product  of 
B  O  and  its  perpendicular  xi/  ? 

81.  The  adjacent  sides  of  a  rhomboid  are   ^'        S~ 
12  inches  and  10   inches   respectively,  and  the  altitude 


108  aUADRILA  TBHALS. 

perpendicular  to  the  longer  8i<le  is  5  inches.     Find  the 
length  of  the  altitude  i)erpendicular  to  the  shorter  side. 

82.  Place  two  eciuilatenil  triangles  so  that  a  side  of  one 
coincides  with  a  side  of  the  other.  The  parallelogram 
thus  formed  is  a  rhombus.  If  a  side  of  the  equilateral  tri- 
angle is  8  inches,  what  is  the  perimeter  of  the  rhombus  ? 

Note.  —  A  rhomlioid  whose  Aides  are  all  equal  is  called  a  Rhombus. 

83.  Draw  two  triangles  each  having  an  angle  of  40° 
included  by  sides  each  6  inches.  Place  the  sides  opposite 
the  given  angles  together,  and  find  the  perimeter  of  the 
rhombus  thus  formed. 

84.  Have  you  ever  seen  panes  of  glass  in  the  shape  of  a 
rhombus  ?  If  so,  where  ?  What  other  name  is  given  to 
that  sha}>e  ? 

85.  Is  the  **•  diamond  **  of  the  baseball  field  a  rhombus 
or  a  square  ? 

86.  How  does  a  rhombus  differ  froiu  a  square  ?  From 
other  rhomboids? 

87.  If  you  have  a  square  frame  4  inches  each  way,  and 
press  ttigether  two  opiK)site  corners,  what  shaj)e  will  the 
frame  t^ike  ?  Will  its  inclosed  space  grow  less  or  remain 
the  same?  Which  is  the  greater,  a  square  whose  sides 
are  each  4  inches,  or  a  rhombus  whose  sides  are  each  4 
inches  ? 

88.  To  what  six  names  is  a  rhombus  entitled  ? 

89.  Draw  a  rhombus  whose  angles  are  equal  to  those 
given  in  Ex.  82.  What  kind  of  triangles  are  formed  by 
its  short  diagonal  ?  How  many  degrees  are  there  in  each 
angle  of  the  triangles  thus  formed  ?  How  many  in  each 
of  the  rhombus  ? 


QUADRILA  TERALS.  109 

90.  What  is  one  side  of  a  rhombus  whose  perimeter  is 
40  centimeters  ? 

91.  Draw  and  cut  out  three  rhombuses  whose  angles 
are  equal  to  those  given  in  Ex.  82.  Place  them  about 
a  common  point,  the  vertex  of  a  large  angle  of  each  coin- 
ciding with  the  vertex  of  a  large  angle  of  the  others. 
WJiat  figure  is  thus  formed  ? 

92.  What  kind  of  a  triangle  is  formed  by  a  diagonal 
and  two  sides  of  a  rhombus  ? 

93.  What  is  the  sum  of  all  the  angles  of  a  rhombus  ? 
Quote  authority. 

94.  How  does  the  number  of  degrees  in  all  the  angles 
of  a  rhombus  compare  with  the  number  in  all  the  angles 
of  a  triangle  ? 

95.  One    side    of    the    rhombus  ^ yB 

ABCB   is  12  centimeters   and  the  y/    \  / 

altitude  AE  is  8  centimeters.     Find         /       \      / 

the  area.  ^  -E    ^ 

96.  What  is  the  area  of  a  rhombus  whose  perimeter 
is  32  centimeters  and  the  altitude  \  of  one  side  ? 

97.  J'ind  the  area  of  a  rhombus  in  which  the  sum  of 
the  altitude  and  one  side  is  19  inches  and  the  perimeter 
is  40  inches. 

98.  Find  the  area  and  perimeter  of  a  rhombus  in  which 
one  side  is  3  inches  more  than  the  altitude,  and  their  sum 
is  15  inches. 

99.  The  area  of  a  rhombus  is  90  square  inches  and  one 
of  its  sides  is  10  inches.  How  far  apart  are  the  parallel 
sides  ? 


110  QUADRILATERALS. 

100.  The  area  of  a  rhombus  is  35  square  centimeters ; 
the  altitude  is  5  centimeters.     Find  the  jjerimeter. 

101.  The  area  of  a  rhombus  is  120  square  inches  and 
its  i>erimeter  is  48  inches.  How  far  apart  are  its  oppo- 
site sides  ? 

102.  Find  one  side  of  a  rhombus  in  which  the  perimeter 
of  the  triangle  formed  by  two  sides  and  the  long  diagonal 
is  22  inches,  the  long  diagonal  being  10  inches. 

103.  Find  the  perimeter  of  a  rhombus  in  which  the  short 
diagonal  is  10  inches  and  the  perimeter  of  one  of  the 
triangles  into  which  it  divides  tlie  rhombus  is  36  inches. 

104.  One  angle  of  a  rhombus  is  60**.  Find  all  the 
others.  Reproduce  the  rhombus  and  draw  its  short 
diagonal.  Into  what  kind  of  triangles  is  the  rhombus 
divided  by  its  short  diagonal  ? 

105.  The  square  A  BCD  is  8  inches  in 
dimensions.  E,  F,  G,  and  JJare  the  mid- 
dle points  of  their  respective  lines.  What 
is  the  area  of  the  square  UFGff? 

SuGOKftTiON. — Draw  construction  Hnes  EG  and 
IIF  and  compare  the  triangles  thus  formed. 

106.  How  long  is  each  diagonal  of  the  inscribed  square  ? 
How  does  the  product  of  the  diagonals  of  the  inscril)ed 
square  compare  with  the  area  of  the  square  in  which  it  is 
inscribed  ? 

107.  Show  that  the  area  of  a  square  is  equal  to  one 
half  the  square  of  one  of  its  diagonals. 

108.  Show  that  the  area  of  a  rhombus  is  equal  to  one 
half  the  product  of  its  diagonals. 


QUADRILA  TERALS.  Ill 

109.  Draw  a  rectangle  6  inches  by  4. 
Connect  the  middle  points  of  the  adjacent 
sides.  How  long  is  the  long  diagonal  of 
the  rhombus  thus  formed?  Its  short 
diagonal?  What  is  the  area  of  the  rec- 
tangle ?  The  area  of  the  rhombus  is  what  fractional  part 
of  the  area  of  the  rectangle  ? 

110.  The  short  diagonal  of  a  rhombus  is  10  inches  and 
the  long  diagonal  is  1^  times  as  long.     Find  the  area. 

111.  One  of  the  diagonals  of  a  rhombus  is  7  inches 
longer  than  the  other,  and  their  sum  is  27  inches.  Find 
the  area. 

112.  The  area  of  a  rhombus  is  144  square  inches.  One 
of  its  diagonals  is  16  inches.     Find  the  other. 

Query.  — What  is  the  area  of  the  rectangle  in  which  the  rhombus 
is  inscribed  ? 

113.  The  perimeter  of  the  triangle  formed*  by  the 
long  diagonal  and  two  sides  of  a  rhombus  is  36  inches. 
The  perimeter  of  the  triangle  formed  by  the  short  diago- 
nal and  two  sides  is  32  inches.  The  perimeter  of  the 
rhombus  is  40  inches.     Find  its  area. 

114.  The  diagonals  of  a  rhombus  are  6  inches  and  8 
inches  and  one  of  its  sides  is  5  inches.  What  is  the  dis- 
tance from  that  side  to  the  opposite  side  ? 

Query.  —  Given  the  area  and  base  of  a  parallelogram,  how  do 
you  find  the  altitude  ? 

115.  Is  every  rhombus  a  rhomboid  ?  Is  every  rhom- 
boid a  rhombus  ? 

116.  Can  you  inscribe  a  rhombus  in  a  circle  ? 


112  QUA  DRILA  TERA  LS. 

117.  Draw  a  square  and  a  rhombus  whose  sides  are 
each  3  inches  long  and  tell  which  is  the  larger. 

Suggestion.  —  Make  the  rhombus  in  such  a  way  that  the  long 
diagonal  will  be  very  long  and  the  short  diagonal  very  short. 

118.  Draw  a  rhombus  whose   base  and  altitude  are 
equal,  if  you  can.     Explain. 

119.  Define  each  term  in  the  following  classification: 
Trapezium. 

Quadrilateral     Trapezoid.         ,  ^  ,        o. 

Parallelogram  I  ^^ctangle  —  Square. 

[  Rhomboid  —  Rhombus : 


CHAPTER   IX. 
RATIO  AND  PROPORTION. 

1.  6  is  how  many  times  3  ?     15  is  how  many  times  5  ? 

2.  What  quotient   is  obtained  by  dividing  28  by  7  ? 
By  dividing  3  by  5  ? 

Note.  —  The    quotient    obtained    by   dividing   one   quantity  by 
another  of  the  same  kind  is  called  their  Ratio. 

3.  What  is  the  ratio  of  10  to  5  ?     Of  21  to  7  ?     Of 
8  to  4  ?     Of  8  to  8  ?     Of  8  to  16  ? 

4.  Complete  the  following  statements  : 

The  ratio  of  10  to  2  is  — .  The  ratio  of  10  to  8  is  — . 

The  ratio  of  10  to  15  is  ~.  The  ratio  of  12  to  8  is  — . 

Note.  —  Instead  of  writing  out  the  whole  statement,  we  use  signs 
to  express  ratio.     10  :  5  =  2  is  read  —  The  ratio  of  10  to  5  is  2. 

Find  the  ratios  of  : 

5.  27:    9  10.  9:    6 

6.  16:    6  11.  10:15 

7.  21:    7  12.  21:28 

8.  5:10  13.    36:24 

9.  8:12  14.    25:15 

15.    '^ If  the  line  m  is  14  feet  long  and  the 

n line  n  is  21  feet  long,  what 

is  the  ratio  of  m  to  n? 

HORN.    GEOM.  —  8  113 


114  RATIO  AND  I'HOPORTION. 

16.  If  the  line  a  is  6   inches  shorter  than  the  line  ft, 
which  is  2  feet  long,  what  is  the  ratio  of  a  to  ft  ? 

17.  Draw  an  arc  of  60°,  and  with  the  same  radius  an 
arc  of  90°,  and  find  the  ratio  of  the  greater  to  the  less. 

18.  What  is  the  ratio  of  a  quadrant  to  the  circum- 
ference of  which  it  is  a  part  ? 

19.  What  is  the  ratio  of  an  angle  of  60°  to  its  comple- 
ment?    To  its  supplement? 

20.  Write  a  ratio  in  which  the  antecedent  is  greater 
than  the  consequent. 

Note.  —  The  first  term  of  a  ratio  i«  called  its  Antecedent;  the 
second  terra  its  Consequent.     Taken  together  they  form  a  Couplet. 

Insert  the  missing  consequent  in  the  following  ratios : 


21.   14:?  =  7. 

27. 

7  :  ?  =  J. 

22.   18:?  =  2. 

28. 

8  :  ?  =  f 

23.    25:?  =  6. 

29. 

7:?  =  J. 

24.    20:?  =  4. 

80. 

10 : ?  =  jf 

26.    28:?  =  4. 

31. 

10  :  ?  =  |. 

26.    35:?  =  7. 

82. 

12  :  ?  =  J. 

8  :  4  =  ?,  6  :  3  =  ? 

33. 

Note.  —  When  the  ratio  of  two  quantities  is  equal  to  the  ratio 
of  two  other  quantities,  the  four  quantities  form  a  Proportion. 
8  :  4  =  6  :  3  is  read  —  The  ratio  of  8  to  4  equals  the  ratio  of  0  to  3. 

34.  Show  that  this  proportion  is  true,  8  :  4  =  10  :  6. 

Query.  — What  is  the  ratio  of  8  to  4?  Of  10  to  5?  How  do  they 
compare? 

35.  Verify  the  proportion  35  :  7  =  15  :  3. 

XoTE,  —  The  first  and  last  terms  of  a  proi>ortion  are  called 
Extremes;  the  second  and  third  are  called  Means.  Do  you  see 
why  they  are  so  called? 


^*«— ^==^^*:R^r/0  AND  PROPORTION.  115 

36.  P^ind   the  last  extreme  of   the   proportion   20  :  10 

=  18  :  ? 

37.  Find  the  fourth  term  of  the  proportion  3:6  =  9:? 

38.  21:7  =  18:? 

39.  4:5  =  12:? 

Suggestion.  —  The  ratio  of  4  to  5  is  f .     12  is  f  of  what  ? 


40.     8 

10  =  24  :  ? 

41.  17 

8  =  85  :  ? 

42.  21 

36=    7:? 

43.  In  the  proportion  8:4  =  6:3,  what  is  the  product 
of  the  extremes  ?  What  is  the  product  of  the  means  ? 
How  do  they  compare  ? 

44.  See  if  the  product  of  the  means  equals  the  product 
of  the  extremes  in  the  proportions  which  you  have  com- 
pleted above. 

Principle  32.  —  In  every  proportion  the  product  of 
the  extremes  is  equal  to  the  product  of  the  means. 

45.  Find  the  value  of  x  in  the  proportion  10  :  5  =  20  :  x 
by  making  a  statement  of  the  equality  between  the  prod- 
uct of  the  means  and  the  product  of  the  extremes,  and 
then  solving  the  equation. 

Find  the  value  of  x  in  the  following  proportions : 

46.  6:  7  =  12:  a;.  49.    5:    2^  =  21:    3. 

47.  4:5=    2^:21.  50.    2^ :    5  =    3  :  15. 

48.  10:2:=    6:    3.  51.    2^:24  =  16:48. 

52.  Write  a  true  proportion  in  which  the  first  antece- 
dent is  20  and  the  second  antecedent  is  40, 


116  RATIO  AND  PROPORTION, 

68.  Write  a  true  proportion  in  which  the  first  antece- 
dent is  greater  than  its  consequent. 

64.  Can  you  write  a  true  proportion  in  which  the  con- 
sequent of  the  first  ratio  is  greater  tlian  its  antecedent, 
and  the  consequent  of  the  second  ratio  is  less  than  its 
antecedent  ?     Give  reasons. 

66.    a  I  h  =  c  I  d,      a    l^eing    a    line   8  _« 

inches,  6,  6  inches,  and  c,  6  inches,  find  fr 

the  length  of  the  line  d.     Construct  the   £ 

rectangle  which   is   the  prcxluct  of  the  ^ 

lines  a  and  d^  and  also  that  which  is  the  product  of  the 
lines  h  and  c.     Compare  their  areas. 

66.  ^  :  i?  =»  (7 :  2>.  If  ^  is  a  line  8  inches,  B,  6  inches, 
and  (7,  4  inches,  how  long  is  D  ?  Find  the  difference  of 
the  areas  of  rectangle  AD  and  rectangle  BC.  Find  the 
difference  of  the  perimeters  of  those  rectangles. 

NoTK.  —  The  use  of  ratio  and  proportion  is  not  confined  to 
geometry.  Com|>ariiig  the  value  of  1  hat  at  $2.00  ^ith  the  vahie  of 
5  hat«  at  the  same  price,  we  nee  that  the  more  hata  we  buy  at  a  certain 
price  tlie  more  money  we  must  pay.     1  hat :  5  hats  =  92.00 :  •  10.00. 

67.  Complete  the  proportion  which  expresses  the  rela- 
tion of  6  hats,  7  hats,  and  their  values,  when  6  hats  are 
worth  |l  16.00.     6  hats  :  7  hats  =  J§i  16.00  :  ? 

68.  Complete  the  proportion,  2  yards  of  silk  :  7  yards 
of  silk  =  f  6.00,  the  price  of  2  yards  :  ? 

59.  If  1  apple  costs  2  cents,  1  apple  :  7  apples  =  2 
cents  :  ?     3  apples  :  8  apples  =  6  cents  :  ? 

60.  What  is  the  ratio  of  an  arc  of  120®  to  the  semicir- 
cumference  of  which  it  is  a  part  ? 

61.  The  line  a  is  8  centimeters.  The  sum  of  the  lines 
a  and  h  is  20  centimeters.     What  is  the  ratio  of  the  less 


RATIO  AND  PROPORTION.  117 

line  to  the  greater  ?     How  many  of  tlie  shorter  lines  equal 
three  of  the  longer  lines  ? 

62.  If  ^CMs  6  inches,  and  ^^  :  5(7=  4  :  3,  how  long 
is  ABl     Find  the  ratio  of  AO  io  BC. 

Of  AB  to  AC,  ^' ^ ^G 

63.  If  AC  is  15  inches,  and  AC'.BC  =  ^:1,  what  is 
the  ratio  oi  AB  to  BC^ 

64.  If  the  arc  AC  is  30  inches,  and  the 
arc  AB  is  20  inches,  what  is  the  ratio  of  arc 
AC  to  arc  5(7?     Of  arc  AB  to  arc  5^? 

Q6.    If  angle  ABC  is  20°,  and  angle  CBD 
is  50°,  what  is  the  ratio  of  the  greater  to 
the  less  ?     Of  the  less  to  the  greater  ?     Of 
the  less  to  their  sum?     Of  the  greater  to      ^     "^~^z) 
their  difference  ?     Find  the  ratio  of  an  angle  of  50°  to  its 
complement.     To  its  supplement.     Find  the  ratio  of  the 
complement  of  an  angle  of  50°  to  its  supplement. 

%Q.  In  the  circle  whose  center  is  0, 
angle  BOC:  angle  C01)  =  S  :  2.  The  arc 
56' is  21  inches.  Find  the  arc  CD.  Arc 
CD :  arc  DU  =  3:1.  How  long  is  the  arc 
5^?  If  the  arc  BC  were  15  inches,  how 
long  would  be  the  arc  BJS? 

67.  a  :  h  =  c  '.  d.  «  =  an  angle  of  40°,  h  an  angle  of 
70°,  c  an  angle  of  80°.  Find  the  number  of  degrees  in 
the  angle  d. 

68.  Given  the  ratio  7:5.  Multiply  both  terms  by  10, 
giving  the  ratio  70  :  50.  Compare  it  with  the  original 
ratio.  Is  it  true  that  70  :  50  =  7  :  5  ?  Can  you  obtain  a 
true  proportion  in  the  same  way  by  using  as  a  multiple 
any  other  number  than  10  ? 


118  RATIO  AND  PROPOHTION, 

69.  Experiment  by  multiplying  both  terms  of  ratios 
by  the  same  number  until  you  are  convinced  of  the  truth 
of  the  following  statement : 

Principle  8^3.  —  Two  quantities  are  in  the  same  ratio 
a^  their  eguimidtiples. 

Query.  —  What  are  equimultiples? 

70.  The  areas  of  two  triangles  are  to  each  other  as 
4  to  5.  What  is  the  ratio  of  the  areas  of  the  rectangles 
whose  bases  and  altitudes  are  respectively  equal  to  those 
of  the  triangles  ?  If  the  area  of  the  smaller  triangle  is  20 
square  inches,  what  is  the  area  of  each  of  the  other  three 
figures  ? 

71.  What  is  the  ratio  of  the  circumferences  of  two 
circles,  the  diameter  of  one  being  7  inches,  and  of  the 
other  14  inches? 

72.  Find  the  ratio  of  the  circumferences  of  two  circles, 
the  radius  of  one  being  20  inches,  and  of  the  other  10 
inches. 

78.  With  radii  of  different  lengths  construct  several 
circles.  Compare  their  circumferences,  and  illustrate  the 
following  principle  : 

Principle  34.  —  TTie  circumferences  of  circles  are  in 
the  same  ratio  to  each  other  as  their  diameters  or  their 
ra^ii, 

74.  Are  the  diameters  of  circles  equimultiples  of  their 
radii?  Are  the  circumferences  equimultiples  of  diam- 
eters?    Of  radii?     Show  the  truth  of  your  answers. 

75.  Two  circumferences  are  in  the  ratio  of  3  to  1.  The 
/liameter  of  the  smaller  circle  is  17  inches.  Find  that  of 
the  greater. 


RATIO  AND  PROPORTION. 


119 


76.  The  radius  of  the  greater  circle  is 
the  diameter  of  the  smaller.  What  is  the 
ratio  of  their  circumferences  ?  If  the  cir- 
cumference of  the  greater  is  6  centimeters, 
what  is  the  circumference  of  the  smaller  ? 

77.  0  is  the  center  of  the  circle  ;  AUG 
and  CFB  are  semicircumferences.     If  the 
circumference  of  the  circle  whose  ceater  is   a 
0  is  30  millimeters,  what  is  the  sum  of  the 
arcs^^(7and(7FJ5? 

78.  Three  equal  semicircles  are  placed 
upon  a  line  AB,  and  the   semicircumfer- 
ence  AMB  is  drawn.     If  the  arc  AMB  is    ^ 
36  inches,  what  is  the  sum  of   the   arcs 
AUG,  GFB,  and  LGBl 

79.  AB  =  BG  =  GB.     If    the    semicir- 
cumference  AUG  is  12  centimeters,  how 
long  is  the  semicircumference  GGB  ?    How    ^ 
long  is  the  semicircumference  AFB  ? 

80.  (7  is  the  center  of  the  circle.  If  its 
circumference  is  40  centimeters,  how  long 

is  the  double  curve  AEGFB^  composed  of  ^ 
two  semicircumferences  ?  The  irregular  fig- 
ure AJEGFBM  is  what  part  of  the  circle  ? 

81.  AB==BG=  GB.     If   the  semicir- 
cumference AFB  is  15  centimeters,  how 
long    is    the    semicircumference     GGB?    ^ 
How  long  is  the  double  curve  composed  of 
the  two  circumferences  AEG  and   GG-D  ? 


120 


liATIO  AND  PROPOIITION. 


82,  The  diameter  AD  is  divided  equally 
at  the  points  B  and  (7,  and  the  double 
curves  are  each  composed  of  semicircura- 
ferences.  If  the  circumference  of  the  cir- 
cle is  30  centimeters,  how  long  is  the 
perimeter  of  the  irregular  figure  AGCHDfBE^ 


83.  The  diameter  AD  is  trisected  at 
the  points  B  and  C.  AD  =  21  inches. 
AEC  and  CHD  are  semicircumferences. 
Find  the  length  of  the  jKjrimeter  of  the 
irregular  figure  AECIJDG. 


84.  Tlie  side  of  one  square  is  3  feet,  of  another  6  feet. 
What  is  the  ratio  of  the  i)erimeters  of  those  squares  ? 

85.  The  side  of  one  square  is  2  feet,  of  another  1  yard. 
Find  the  ratio  of  their  perimeters. 

8G.  Find  the  ratio  of  the  perimeters  of  two  equilateral 
triangles,  a  side  of  one  being  8  inches  and  of  the  other  10 
inches. 

87.  Find  the  ratio  of  the  perimeters  of  two  regular 
hexagons,  a  side  of  one  being  4  inches  and  of  the  other  7 
inches. 

88.  Draw  two  squares,  a  side  of  one  being  twice  as 
long  as  a  side  of  the  other,  and  find  the  ratio  of  their 
areas. 

89.  Find  the  ratio  of  the  areas  of  two  squares,  a  side  of 
one  being  3  times  as  long  as  a  side  of  the  other. 

90.  Find  the  ratio  of  the  areas  of  two  squares  whose 
sides  are  in  the  ratio  of  4:1. 


RATIO  AND  PROPORTION.  121 

91.  What  is  the  perimeter  of  a  square  each  side  of 
which  is  3  times  as  long  as  the  side  of  another  square 
whose  perimeter  is  20  inches  ? 

92.  The  equilateral  triangle  ABQ  has 
each  side  3  tiijies  as  long  as  the  homolo- 
gous side  of  the  equilateral  triangle  ADE. 
What  is  the  ratio  of  their  areas  ? 

Suggestion.  —  Reproduce  and  fold  the  larger 
triangle  according  to  the  indicated  lines. 

93.  Build  from  inch  cubes  a  cube  2  inches  each  way. 
How  many  inch  cubes  make  the  cube  ?  Build  a  ^-inch 
cube.  How  many  layers  of  inch  cubes  are  there  ?  How 
many  inch  cubes  are  in  each  layer  ?  Find  the  ratio  of  the 
volumes  or  contents  of  a  2-inch  cube  and  a  3-inch  cube. 

94.  Find  the  ratio  of  the  volumes  of  a  3-inch  cube  and 
a  4-inch  cube.     Find  the  ratio  of  their  surfaces. 

95.  Find  the  ratio  of  a  cube  whose  edge  is  2  inches  to 
one  whose  edge  is  4  inches. 

96.  Find  the  ratio  of  a  cube,  one  of  whose  faces  con- 
tains 9  square  inches,  to  a  cube,  each  of  wliose  faces  con- 
tains 16  square  inches. 

97.  Find  the  ratio  of  a  cube,  the  sum  of  whose  edges  is 
24  inches,  to  that  of  a  cube,  the  sum  of  whose  edges 
is  60  inches. 

Query.  —  How  many  edges  has  a  cube  ? 

98.  Find  the  ratio  of  a  parallelopiped  8  inches  long, 

5  inches  wide,  and  4  inches  deep  to  one  10  inches  long, 

6  inches  wide,  and  2  inches  deep. 


122  RATIO  AND  PROPORTION. 

99.  A  box  9  inclies  long,  7  inches  wide,  and  3  inches 
deep  will  contain  how  many  times  as  much  as  a  box  3 
inches  long,  1  inch  wide,  and  3  inches  deep  ? 

100.  Find  the  ratio  of  the  areas  of  two  rectangles,  one 
being  8  centimeters  long  and  5  centimeters  wide,  the 
other  being  10  centimeters  long  and  6  centimeters  wide. 

101.  Draw  rectangles  of  different  dimensions  and  illus- 
trate by  numbers  the  truth  of  the  following  principle: 

Principle  35.  —  The  areas  of  any  two  rectangles  are 
in  the  same  ratio  as  tlie  products  of  their  bases  by  their 
altitudes, 

102.  Solve  the  proportion  8x4:6x4  =  8:? 

103.  AB  =  7   inches;    5C  =  4  ^^ 
inches  ;  AF  =  3  inches.     Find  the 
ratio  of  the  rectangles  ABEF  and 
BODE.   Of  ACDFmuX  BODE.        '                  e  " 

104.  Draw  rectangles  with  equal  altitudes  and  bases  of 
different  lengths  and  illustrate  the  following  principle  : 

Principle  36.  —  Two  rectangles  having  equal  alti- 
tudes are  proportional  to  their  bases. 

105.  Draw  rectangles  having  equal  bases  and  different 
altitudes  and  illustrate  the  following  principle : 

Principle  37.  —  Two  rectangles  having  equal  ba^es 
are  proportional  to  their  altitv^les. 

106.  What  is  the  ratio  of  the  area  of  a  triangle  whose 
base  is  10  centimeters  and  altitude  8  centimeters  to  that 
of  a  triangle  whose  base  is  12  centimeters  and  altitude 
5  centimeters  ? 


RATIO  AND  PROPORTION.  123 

107.  Triangle  ABiJ  has  a  base  10  inches,  altitude  7 
inches.  Triangle  DEF  has  a  base  5  inches,  altitude 
10  inches.     Find  the  ratio .  of  their  areas. 

108.  Compare  areas  of  triangles  until  you  are  able  to 
complete  the  statement ; 

The  areas  of  two  triangles  are  to  each  other  as  the 
products  of ? 

109.  Find  the  areas  of  triangles  having  equal  bases 
and  differing  altitudes  until  you  can  complete  the  state- 
ment: 

Two  triangles  with  equal  bases  are  to  each  other  as ? 

110.  Find  the  areas  of  triangles  having  equal  altitudes 
and  differing  bases  until  you  can  complete  the  state- 
ment: 

Two  triangles  having  equal  altitudes  are  to  each  other 
as ? 

111.  Reproduce  the  triangle  ABO^  right-angled  at  B. 
If  AB  is  8  inches  and  BO  is  6  inches,  AC  will 

be  10  inches  (a  fact  which  you  will  soon  learn  to 
prove).  At  i>,  the  middle  point  of  AB,  draw 
BE  parallel  to  the  base.  Fold  the  triangle 
along  the  dotted  lines,  and  show  that  the  four 
triangles  thus  formed  are  equal.  What  is  the 
ratio  of  AB  to  AB  ?    Of  AE  to  AO?    Of  BE  to  BO? 

112.  Prove  by  reference  to  a  geometric  principle  that 
angle  ABE=  angle  AB  0.    That  angle  AEB  =  angle  A  OB. 

113.  Draw  a  triangle  one  of  whose  sides  (not  a  base) 
is  12  centimeters.  At  different  points  in  the  side  draw 
lines  parallel  to  the  base,  measure,  and  illustrate  the  fol- 
lowing statement : 


124 


RATIO  AND  PROPORTION, 


Principle  38.  —  A  line  drawn  parallel  to  the  base 
of  a  triangle  divides  the  sides  proportionally. 

114.  EF  and  DGr  are  jmrallel  to  ^ 
BC. 

AB  =  6  inches.     AC  =12  inches. 
AE=^  inches.    ED=    2  inches. 
Find  ^jP,  J^G^,  and  GO. 

115.  AB  =  \b  centimeters. 
^(7=  18  centimeters. 
DE  is  parallel  to  A  (7. 
AD  =  5  centimeters. 

Find  EC  and  EB, 


CHAPTER    X. 

CUMULATIVE    REVIEW   No.  3. 


mmm 


1.  The  above  is  one  pattern  of  an  ornamental  border 
called  the  "Greek  Fret."  How  many  right  angles  are 
there  in  the  piece  here  represented  ? 

2.  Draw  a  rectangle  which  is  the  product  of  a  line 
9  inches  and  a  line  5  inches,  and  a  rhomboid  which  lias 
the  same  area.     Which  has  the  greater  perimeter  ? 

3.  Lay  off  a  line  5  inches  on  a  line  8  inches,  and  con- 
struct the  square  of  their  difference. 

4.  Cut  out  the  square  of  a  line  6  inches  from  one 
corner  of  the  square  of  a  line  8  inches,  and  find  the  area 
and  perimeter  of  the  irregular  hexagon  which  is  their 
difference. 

5.  How  many  times  can  the  G.  C.  M.  of  a  line  35  inches 
and  one  20  inches  be  laid  off  on  the  shorter  line  ? 

6.  How  many  times  can  the  longer  of  two  lines,  one 
8  inches  and  the  other  12  inches,  be  laid  off  on  their 
L.  C.  M.  ? 

7.  Is  the  smaller  square  inscribed  in  the 
greater  ?     Give  reasons  for  your  answer. 


125 


I2e  CUMULATIVE  REVIEW, 


H 


8.  A,  0,  E,  and  G-  are  the  middle 
points  of  their  respective  lines.  Repro- 
duce and  show  what  part  the  inscribed 
square  is  of  the  outer  square. 

'  E 

9.  If  the  side  HB  is  8  inches,  what  is  the  area  of 
the  inscribed  square  ?  If  AB  equals  5  inches,  what  is 
the  area  of  the  inscribed  square? 

10.  Find  the  area  and  perimeter  of  the  rectangle 
whose  longer  sides  are  each  the  sum  of  two  lines,  11 
centimeters  and  4  centimeters,  and  whose  shorter  sides 
are  each  the  diiference  of  those  lines. 

11.  Find  the  area  and  perimeter  of  the  rectangle 
which  is  the  product  of  the  sum  and  the  difference  of 
two  lines,  one  12  inches,  the  other  8  inches. 

12.  Three  arcs  compose  a  circumference.  The  first  is 
6  times  the  second,  and  the  third  is  4  times  the  second. 
How  many  degrees  are  there  in  each  ?  How  long  is  each 
arc  if  the  radius  is  2  feet  11  inches  ? 

13.  How  many  millimeters  are  there  in  the  circum- 
ference of  a  circle  whose  radius  is  3J  centimeters  ? 

14.  A  degree  of  a  circumference  whose  radius  is  21 
inches  is  what  fractional  part  of  a  degree  of  a  circum- 
ference whose  radius  is  42  inches  ? 

15.  How  many  degrees  are  there  in  an  arc  which  con- 
tains 30°  more  than  its  supplement  ? 

Query.  — Will  you  let  x  =  the  arc  or  the  supplement? 

16.  How  long  is  an  arc  of  72°  of  a  circumference  whose 
diameter  is  35  millimeters  ? 


CUMULATIVE   REVIEW.  127 

17.  Find  the  perimeter  of  a  sector  which  is  -^  of  a 
circle  whose  diameter  is  17^  inches. 

18.  Make  ten  angles  of  36  each  around  a  common  point 
by  lines  5  inches  long.  Join  the  extremity  of  each  line 
to  the  extremities  of  the  neighboring  lines.  Is  the  poly- 
gon thus  formed  a  regular  polygon  ?  Write  its  name 
upon  it. 

Note.  —  A  polygon  of  ten  sides  is  called  a  Decagon. 

19.  Divide  the  circumference  of  a  circle  whose  diam- 
eter is  8.4  inches  into  two  arcs,  one  of  which  is  7  times 
the  other. 

20.  What  is  the  ratio  of  a  circumference  to  an  arc 
whose  remaining  arc  is  288°  greater  ? 

21.  Divide  a  circle  into  sectors  whose  arcs  are  120°  (by 
making  central  angles  with  a  protractor),  and  show  the 
ratio  of  each  sector  to  the  circle. 

22.  Angle  h  =  angle  a  +  40°.  Find  all 
the  angles  formed  by  the  transversal  and 
the  parallels. 

23.  How  many  degrees  are  there  in  each  of  the  three 
angles  formed  at  a  given  point  on  the  same  side  of  a 
straight  line  if  the  first  is  twice  the  second  and  the  second 
is  3  times  the  third  ? 

Suggestion. —  a:  =  ? 

24.  The  shortest  side  of  a  scalene  triangle  is  15  inches. 
Another  side  is  33J%  longer  than  the  first.  The  third 
side  is  25%  longer  than  the  second.  Find  the  perimeter. 
Find  the  ratio  of  the  shortest  side  to  the  longest. 


128  CUMULATIVE  REVIEW. 

25.  One  side  of  a  scalene  triangle  whose  perimeter  is 
142  decimeters  is  17  decimeters  longer  than  one  of  its 
adjacent  sides  and  21  decimeters  longer  than  the  other 
side.     What  is  its  ratio  to  the  perimeter  ? 

26.  Each  of  the  equal  sides  of  an  isosceles  triangle  is  18 
inches.  The  base  is  33^%  of  one  of  the  equal  sides.  Find  the 
perimeter.    What  is  the  ratio  of  the  perimeter  to  the  base  ? 

27.  Find  the  perimeter  of  the  isosceles  triangle  whose 
base  is  14  inches,  and  whose  equal  sides  are  each  25% 
longer  than  the  base. 

28.  Find  the  area  of  an  isosceles  triangle  whose  perim- 
eter is  32  inches,  each  of  whose  equal  sides  is  12 J  inches, 
and  whose  altitude  is  12  inches. 

29.  An  exterior  vertical  angle  of  an  isosceles  triangle 
is  7  times  the  vertical  angle.  Find  each  angle  of  the 
triangle  and  the  ratio  of  the  base  angle  to  the  vertical 
angle. 

30.  Construct  an  equilateral  triangle  each  of  whose 
sides  is  5  inches,  and  upon  each  side  construct  an  equi- 
lateral triangle  and  find  the  perimeter  of  the  figure  thus 
formed.  Write  the  name  of  the  figure.  The  original 
triangle  is  what  fractional  part  of  it  ? 

31.  Each  of  the  base  angles  of  an  isosceles  triangle  is  3 
times  the  vertical  angle.  How  many  degrees  are  there  in 
each  angle  of  the  triangle  ? 

32.  In  a  right-angled  scalene  triangle  the  larger  acute 
angle  is  10  times  the  smaller.  Find  the  number  of  degrees 
in  each  angle. 

33.  How  many  degrees  are  there  in  two  consecutive 
angles  of  a  rhomboid  ? 


CUMULATIVE  REVIEW.  129 

34.  How  many  times  will  a  square  whose  side  is  2 
inches  be  contained  in  a  square  whose  side  is  12  inches  ? 

35.  An  angle  of  a  rhomboid  is  5  times  an  adjacent 
angle.     How  many  degrees  is  each  angle  of  the  rhomboid? 

36.  The  rhomboid  A  BOD  is  78  inches  in  perimeter, 
and  the  sum  of  the  longer  sides  is  twice  the  sum  of  the 
shorter  sides.     Find  the  ratio  of  two  adjacent  sides. 

37.  Find  the  area  of  a  rhomboid  whose  perimeter  is  40 
inches,  its  shorter  sides  being  each  7  inches,  and  the  per- 
pendicular distance  between  the  longer  sides  6  inches. 

38.  When  all  the  sides  of  a  rhomboid  are  equal,  what 
is  the  figure  called  ?  If  the  perimeter  of  a  rhombus  is  20 
inches,  what  is  one  side  ? 

39.  What  is  the  ratio  of  the  area  of  a  rhombus  to  that 
of  the  rectangle  of  its  diagonals  ?     Illustrate  by  diagram. 

40.  Find  the  area  of  a  rhombus,  one  of  whose  diagonals 
is  6  inches  longer  than  the  other,  and  their  sum,  20  inches. 

41.  Draw  two  triangles  having  two  sides  and  the 
included  angle  of  one  respectively  equal  to  two  sides  and 
the  included  angle  of  the  other,  and  compare  their  areas. 

42.  AD  =  24  inches  ,  BC=  20  inches  ;  AB 
=  26  inches.  AD  is  perpendicular  to  J5C  at 
its  middle  point.  Find  the  perimeter  and  area 
of  the  triangles  ^i>(7  and  ABC 

43.  Draw  two  lines,  8  inches  and  6  inches,  ^ 
cutting  each  other  perpendicularly  at  their  middle  points. 
Join  each  extremity  of  the  lines  to  the  extremity  of  its 
neighboring   line.      Can  you  prove  that  the  figure  thus 

HORX.    GEOM. 9 


130 


CUMULATIVE  REVIEW. 


formed  is  a  rhombus? 
which  apply. 


Quote  the  geometric  principles 


44.  AB  =  10  inches  ;  i>(7  =  12 
inches  ;  the  altitude  =  5  inches  ;  find 
the  area  of  the  trapezoid. 


45.  The  trapezium  ABOD  has 
the  side  AB  7  inches  longer  than 
the  side  AD^  and  7  inclies  less 
than  the  side  BQ  \  DC  in  1  inches 
longer  than  the  side  BC;  tlie  d 
perimeter  is  70  inches.  What  is  the  ratio  of  the  side  AD 
to  the  perimeter  ? 

46.  The  side  AD  of  the  trapezium  A  BCD  is  5  ^ 
centimeters  longer  than  DC  ;  DC  is  4  centimeters 
longer  than  BC;  BC  is  3  centimeters  longer  than 
AB  ;   the  perimeter  is  50  centimeters.    Find  each 
side. 

47.  Can  you  draw  a  trapezium  such  that  one  of  its 
diagonals  will  divide  it  into  two  equal  triangles  ? 

Suggestion.  —  Kite. 

48.  When  a  carriage  is  moving,  which  moves  in  space 
the  more  rapidly,  a  point  on  the  liub  of  a  wheel,  or  a  point 
on  the  tire  ?     Draw  curves  showing  the  path  of  each. 

49.  The  diameter  ^^=14  inches; 
CD  =  21  inches  ;  angle  UOD  =  40°  ; 
find  the  length  of  arc  UD  and  arc  FB. 
Compare  the  ratio  of  the  arcs  with  the 
ratio  of  the  circumferences.  With  the 
ratio  of  the  diameters. 


CUMULATIVE  REVIEW.  131 

50.  If  the  diameter  AB  were  10  inches,  CD  15  inches, 
and  the  arc  FB  were  8  inches,  what  would  be  the  length 
of  the  arc  jE^i)  ? 

51.  If  OB  were  9  inches,  OB  15  inches,  the  arc  ED  12 
inches,  how  long  would  the  arc  FB  be  ? 

52.  If  the  circumference  of  the  larger  circle  were  50 
inches,  that  of  the  smaller  30  inches,  and  the  arc  EB  5 
inches,  what  would  be  the  length  of  the  arc  FB"^ 

53.  Cut  out  six  squares,  the  dimensions  of  each  being  1 
decimeter,  and  fasten  them  together  (by  gluing  or  sewing) 
so  as  to  make  a  cubic  decimeter.  How  many  centimeters 
measure  the  length  of  all  the  edges  of  the  figure  ? 

Note.  —  A  cubic  decimeter  is  called  a  Liter. 

54.  How  many  square  centimeters  are  there  in  all  the 
faces  of  a  liter  ?  How  many  in  all  the  squares  that  can 
be  inscribed  in  the  faces  of  a  liter  ? 

^t).  Draw  a  square  decimeter,  and  at  the  middle  point 
of  each  side  draw  a  perpendicular  2  decimeters  long. 
From  the  extremity  of  each  perpendicular  draw  oblique 
lines  to  the  side  of  the  decimeter  upon  which  the  perpen- 
dicular stands.     Find  the  area  of  the  polygon  thus  formed. 

bQ.  Cut  out  the  figure  described  in  Ex.  55,  and  fold 
the  four  triangles  back  so  that  their  vertices  meet  in  a 
common  point,  and  they  form  a  rectangular  pyramid. 

Find  the  area  of  the  lateral  or  side  surfaces 
of  the  pyramid.     Of  all  its  surfaces. 

57.  Place  this  pyramid  upon  a  liter  so  that 
its  base  coincides  with  a  side  of  the  liter,  and 
find  the  lateral  surfaces  of  the  solid  thus  formed. 


CHAPTER  XL 

POLYGONS. 

XoTE.  — A  Polygon  is  a  plane  figure  bounded  by  straight  lines. 
A  Regular  Polygon  is  one  wliicli  luis  all  its  sides  and  all  iU  angles 
equal.  A  polygon  of  three  sides  is  called  a  Triangle ;  of  four  sides  a 
Quadrilateral ;  of  five  sides  a  Pentagon ;  of  six  sitles  a  Hexagon ;  of 
seven  sides  a  Heptagon ;  of  eight  sides  an  Octagon ;  of  nine  sides  a 
Nonagon ;  of  ten  sides  a  Decagon ;  of  twelve  sides  a  Dodecagon. 

1.  What  plane  tigure  have  you  studied  that  is  not  a 

polygon  ? 

2.  Construct  a  regular  polygon  of  three  sides,  each  of 
which  is  8  centimeters,  and  write  its  name  upon  it. 

3.  Draw  a  regular  polygon  of  four  sides  and  write  its 
name  upon  it. 

4.  Make  five  equal  angles  around  a  common  point  by 
equal  lines,  and  join  the  extremity  of  each  line  to  the 
extremities  of  the  adjacent  lines.     Can  you  prove  that 

the  joining  lines  are  equal  ? 

Suggestion.  —  See  Geom.  Prin.  24. 

The  surface  inclosed  by  the  joining  lines  forms  what 
kind  of  a  polygon  ? 

5.  Draw  a  circle  whose  radius  is  one  of  the  equal  lines 
which  form  the  angles  at  the  center  of  the  figure  which 
you  drew  for  Ex.  4.  Will  it  touch  the  vertices  of  all  the 
angles  of  the  polygon  ?     Argue  for  your  statement. 

132 


POLYGONS.  133 

6.  Reproduce  the  pentagon  whose  center  is  0,  the 
angles  at  0  being  equal,  and  formed  by- 
equal  lines.  What  kind  of  a  triangle  is 
OBC^  OB  A?  Are  they  equal  triangles? 
Why  ?  How  many  degrees  has  each  angle 
of  the  triangle  A  OEl  How  many  degrees 
has  angle  EAB^  Angle  ABC^  Each  an- 
gle of  the  pentagon  ?     Is  it  a  regular  pentagon  ? 

7.  Reproduce  the  figure  in  Ex.  6,  and  draw  lines  from 
the  center  to  the  middle  point  of  each  side.  Can  you 
prove  those  lines  to  be  perpendicular  to  the  sides  ?  Are 
they  equal  ? 

Note.  —  A  line  drawn  from  the  center  of  a  regular  polygon  to  the 
middle  point  of  one  of  its  sides  is  called  its  Apothem. 

8.  With  an  apothem  as  a  radius,  describe  a  circumfer- 
ence. Will  it  include  the  extremities  of  all  lines  drawn 
from  the  center  to  the  middle  point  of  a  side?  Prove. 
Are  the  sides  of  the  polygon  tangent  to  the  circle  ? 

9.  Show  how  you  inscribe  a  circle  in  a  regular  polygon. 

10.  Place  six  equal  equilateral  triangles  around  the 

common  point  B.     With  B  as  a  center,  - a 

and  a  radius  equal   to  a   side  of  one  of     ,/  \  /  \\ 

the  triangles,  circumscribe  a  circumference     jh ^ 4c 

about   the   hexagon   thus   formed.      How     '\   /  \    /' 
may  you  know  that  each  vertex  will  be  -— -^ 

in  the  circumference  ? 

11.  Which  is  longer,  BA  or  ^(7?     Give  reason. 

12.  How  does  the  side  of  a  regular  hexagon  compare 
with  the  radius  of  the  circle  in  which  the  hexagon  is 
inscribed  ? 


134 


POLYGONS. 


Principle  39.  —  The  side  of  a  regular  hexagon  is 
equal  to  the  radius  of  the  circle  in  which  it  is  inscribed. 

13.  How  many  times  does  the  side  of  a  regular  hexagon 
lie  as  a  chord  around  the  circumscribed  circumference  ? 

14.  Draw  a  circle  with  a  radius  of  6  inches  and  inscribe 

a  regular  hexagon. 

15.  Inscribe  a  regular  hexagon  in  a  circle  whose  radius 
is  7  inches.  Join  the  alternate  vertices  with  the  center, 
forming  three  polygons.  What  kind  of  polygons  are 
they  ?     How  long  is  the  perimeter  of  each  ? 

16.  The  sum  of  the  tliree  radii  bears  what  ratio  to  the 
perimeter  of  the  hexagon?  The  perimeter  of  each  rliom- 
bus  bears  what  ratio  to  the  perimeter  of  the  hexagon  ? 

17.  ABCBEF  is  a  regular  hexagon.     Which  is 


greater,  the  triangle  BOO  ov  the  triangle 
OBE^  Give  reasons.  Can  you  prove 
that  OH,  the  altitude  of  the  triangle  OOE, 
equals  HB,  the  altitude  of  the  triangle 
CBE"^  If  the  radius  of  the  circle  is  12 
inches,  how  long  is  OHf    AH> 

18.  Construct  an  equilateral  triangle 
upon  one  side  of  a  regular  hexagon.  The 
area  of  the  irregular  pentagon  thus  formed 
bears  what  ratio  to  that  of  the  hexagon  ? 
What  is  the  ratio  of  their  perimeters  ? 

19.  Construct  an  equilateral  triangle 
upon  each  side  of  a  regular  hexagon.  The 
area  of  the  six-pointed  star  tlits  formed 
bears  what  ratio  to  the  hexagon?  What 
is  the  ratio  of  their  perimeters  ? 


the 


POLYGONS,  135 

20.  Inscribe  the  regular  hexagon  ABCDEF  in  the 
circle  whose  center  is  0,  Join  the  alter- 
nate vertices,  making  the  triangle  BFD, 
How  many  degrees  are  there  in  each  arc 
subtended  by  the  side  of  the  hexagon? 
How  many  degrees  are  there  in  each  arc 
subtended  by  a  side  of  the  triangle  BFD! 
The  triangle  BCD  is  what  fractional  part  of  the  rhombus 
BOD 01  The  triangle  BFD  is  what  fractional  part  of 
the  hexagon  ?  The  line  FII  is  what  fractional  part 
of  the  line  FO"!  Considering  BD  the  base  of  the  tri- 
angle BFD,  what  is  its  altitude?  What  is  the  ratio 
of  the  altitude  of  an  equilateral  triangle  to  the  diameter 
of  the  circle  in  which  it  is  inscribed  ? 

21.  What  is  the  length  of  an  arc  of  60°  in  a  circle 
whose  radius  is  70  inches?  How  long  is  the  chord  which 
subtends  that  arc  ? 

22.  How  long  is  the  perimeter  of  a  segment  whose  arc 
is  60°  of  a  circle  whose  radius  is  S^  feet? 

23.  Find  the  difference  between  the  straight  boundary 
and  the  curved  boundary  of  a  segment  whose  arc  is  60° 
of  a  circle  whose  radius  is  10  feet. 

24.  If  the  base  of  a  triangle  is  10  feet  and  its  altitude 
is  8.7  feet  nearly,  what  is  the  approximate  area  of  the 
triangle  ? 

25.  Inscribe  a  regular  hexagon  in  a  circle.     Cut  the 


hexagon  into  triangles.  Show  tliat  the  area  of  the  hex- 
agon is  equal  to  one  half  the  sum  of  all  its  sides  multi- 
plied by  its  apothem. 


136 


POLYGONS. 


Peinciple  40.  —  TJie  area  of  a  regular  polygon  is 
equal  to  one  half  the  perbrveter  viultiplied  by  the 
apothem. 

26.  A  side  of  a  regular  hexagon  is  8  inches  and  the 
apothem  is  6.9  +  inches.     Find  the  approximate  area. 

27.  A  side  of  a  regular  pentagon  is  6  inches  and  the 
apothem  is  4.12  +  inches.     Find  the  area. 

28.  Find  the  area  of  the  square  ABCD 
each  of  whose  sides  is  10  inclies,  by  divid- 
ing it  into  triangles.  How  long  is  the 
apothem  EFl 

29.  With  a  radius  equal  to  one  half  the 
diagonal  of  a  square,  describe  a  circumfer- 
ence. Show  that  the  square  is  inscribed 
within  it.  Bisect  each  side  of  the  square  by 
a  radius  and  join  the  extremity  of  each 
radius  with  the  adjoining  vertices  of  tlie  square, 
kind  of  a  polygon  is  formed  ? 

30.  A  side  of  a  regular  octagon  being  7  inches  and  the 
apothem  8.44  +  inches,  wliat  is  the  area? 

31.  Inscribe  a  regular  hexagon  in  a  circle,  bisect  each 
arc,  and  join  the  points  of  bisection  with  the  vertices  of 
the  adjoining  angles  of  the  hexagon.  Wliat  kind  of  a 
polygon  is  thus  formed  ? 

32.  The  side  of  a  regular  dodecagon  is  11  inches  and 
the  apothem  is  20.52  -f  inches.      Find  its  area. 

33.  Show  how  you  inscribe  a  regular  dodecagon  in  a 

circle. 

34.  A  side  of  a  regular  octagon  is  9  inches  and  the 
apothem  is  10.86  +  inches.     Find  its  area. 


What 


POLYGONS.  187 

35.  Show  how  you  inscribe  a  regular  octagon  in  a 
circle. 

36.  Draw  a  pentagon,  as  in  the  figure,  and  draw  as 
many  diagonals  as  possible  from  one  of  the 

vertices.  Into  how  many  triangles  is  the 
pentagon  divided?  What  is  the  sum  of  all 
the  angles  of  the  triangles  ?  Then  how  many 
degrees  have  all  the  angles  of  the  pentagon  ? 

37.  Divide  a  hexagon  in  the  same  way  and  show  how 
many  degrees  there  are  in  the  sum  of  all  its  angles. 

38.  Discover  how  many  degrees  there  are  in  the  sum 
of  all  the  angles  of  a  heptagon. 

39.  Discover  the  sum  of  all  the  angles  in  an  octagon. 

40.  Have  you  found  out  the  law  by  which  we  can 
compute  the  sum  of  all  the  angles  of  a  polygon  of  any 
number  of  sides  ? 

41.  How  does  the  number  of  sides  of  a  polygon  com- 
pare with  the  number  of  triangles  into  which  it  is  divided 
by  the  diagonals  drawn  from  a  single  vertex  ? 

Principle  41.  —  The  sum  of  all  tJie  angles  of  a  poly- 
gon is  equal  to  two  right  angles  taken  two  less  times 
than  the  polygon  has  sides. 

42.  How  many  degrees  are  there  in  all  the  angles  of 
a  dodecagon  ?     Of  a  decagon  ? 

43.  How  many  right  angles  are  there  in  the  sum  of  the 
angles  of  a  pentagon  ?     Of  a  hexagon  ?     Of  an  octagon  ? 

44.  How  many  degrees  are  there  in  each  angle  of  a 
regular  hexagon  ?  Of  a  regular  decagon  ?  Of  a  regular 
dodecagon  ? 


;^38  POLYGONS. 

45.  Find  the  number  of  degrees  in 
each  angle  of  the  five-pointed  star 
formed  by  prolongijig  the  sides  of  a 
regular  pentagon  until  they  meet  ? 

46.  Find  the  number  of  degrees  in 
each    angle    of    the    six-pointed    star 
formed  by  prolonging  the  sides   of   a   regular  hexagon 
until  they  meet. 

47.  Find  the  sum  of  the  angles  of  a  polygon  of  24 
sides. 

If  n  represents  the  number  of  sides  of  a  polygon,  what 
represents  the  number  of  right  angles  in  the  sum  of  the 
angles  of  that  polygon  ? 

Find  the  value  of  n  in  the  equation  2  (n  —  2)  =  36. 

Suggestion.  —  2(n  —  2)  =  2n  —  4. 

48.  How  many  sides  has  a  polygon  the  sum  of  whose 

angles  is  36  right  angles  ? 

49.  How  many  sides  has  a  polygon  the  sum  of  whose 

angles  is  2340°  ? 

50.  How  many  sides  has  a  polygon  the  sum  of  whose 
angles  is  1260°  ? 

51.  How  many  sides  has  the  polygon  the  sum  of  whose 
angles  is  twice  the  sum  of  the  angles  of  a  hexagon  ? 

52.  In  what  regular  polygon  is  each  angle  one  half  as 
great  as  an  angle  of  a  regular  hexagon  ? 

53.  Which  have  the  larger  angles,  regular  polygons  of 

few  sides  or  of  many  sides  ? 

54.  Find  the  ratio  of  the  number  of  degrees  in  each 
angle  of  a  regular  hexagon  to  the  number  in  each  angle 

of  a  regular  dodecagon. 


POLYGONS.  139 

Sb.  Find  the  ratio  of  an  angle  of  a  regular  pentagon 
to  an  angle  of  a  regular  decagon. 

56.  Find  the  ratio  of  an  angle  of  a  regular  decagon  to 
an  angle  of  a  regular  dodecagon. 

57.  Find  the  ratio  of  the  sum  of  the  angles  of  a  hexa- 
gon to  the  sum  of  the  angles  of  an  octagon. 

58.  In  what  polygon  is  the  sum  of  the  angles  three 
times  as  great  as  the  sum  of  the  angles  of  a  trapezium  ? 

59.  Prolong  each  side  of  a  square  as  in 
the  figure.  What  is  the  sum  of  all  the 
exterior  angles  thus  formed  ? 

60.  How  many  degrees  has  each  angle  of 
a  regular   pentagon  ?     How  many  degrees 
has  each  exterior  angle  ?    Find  the  sum  of  all  the  exterior 
angles. 

61.  Prolong  each  side  of  a  regular  hexagon  and  dis- 
cover how  many  degrees  there  are  in  the  sum  of  all  its 
exterior  angles. 

62.  Find  in  the  same  way  the  sum  of  all  the  exterior 
angles  of  a  regular  decagon.     Of  a  regular  dodecagon. 

63.  Draw  an  irregular  quadrilateral  and  find  how 
many  degrees  there  are  in  its  exterior  angles. 

64.  Draw  an  irregular  polygon  of  three  sides  and  find 
how  many  degrees  there  are  in  the  sum  of  all  its  exterior 
angles. 

^b.  How  many  degrees  are  there  in 
the  sum  of  the  interior  angle  a  and  the 
exterior  angle  5?  What  is  the  sum  of 
each  pair  of  adjacent  exterior  and  interior 
angles  of  a  pentagon?  Of  a  hexagon? 
Of  an  octagon  ? 


140 


POLYGOXS. 


66.  If  we  let  n  =  the  number  of  sides  of  a  polygon,  is 
the  following  reasoning  coiTect  ? 

Interior  angles  =  lSOn°  -  360^  (Prin.  41.) 

Transposing,  we  have 

Interior  angles  +  360°  =  180  n°. 

Interior  angles  -h  exterior  angles  =  180  n°.       (Prin.  8.) 

Since  if  either  the  exterior  angles  or  360°  is  added  to 
the  interior  angles  the  result  is  the  same,  180  w°,  the 
exterior  angles  must  equal  360°. 

Principle  42.  —  T?ie  suvv  of  all  the  exterior  angles  of 
a  polygon  is  equal  to  four  right  angles. 

67.  What  is  the  ratio  of  the  sum  of  the  interior  angles 
of  an  octagon  to  the  sum  of  its  exterior  angles  ? 

68.  Find  the  ratio  of  the  sum  of  the  exterior  angles  of 
a  decagon  to  the  sum  of  its  interior  angles. 

69.  Find  the  ratio  of  an  exterior  angle  of  a  regular 
dodecagon  to  its  adjacent  interior  angle. 

70.  What  polygon  has  the  sum  of  its  exterior  angles 

equal  to  the  sum  of  its  interior  angles? 

71.  Draw  two  polygons  differing  in  shape  in  which 
each  of  the  interior  angles  is  equal  to  the  adjacent  exterior 
angle. 

72.  What  polygon  has  the  sum  of  its  interior  angles 
equal  to  twice  the  sum  of  its  exterior  angles? 

73.  What  polygon  has  each  of  its  interior  angles  equal 
to  twice  the  adjacent  exterior  angle  ? 

74.  What  polygon  has  the  sum  of  its  exterior  angles 
equal  to  twice  the  sum  of  its  interior  angles  ? 

75.  Of  what  polygon  is  each  of  the  exterior  angles 
equal  to  twice  the  adjacent  interior  angle? 


POLYGONS.  141 

76.  Of  what  polygon  is  each  exterior  angle  equal  to 
two  thirds  of  its  adjacent  interior  angle  ? 

77.  Prolong  the  sides  of  a  regular  octagon  until  they 
meet,  forming  a  star.  Find  the  sum  of  all  the  angles  in 
the  points  of  the  star. 

78.  Can  you  make  a  star  of  a  square  in  the  same  way? 
Give  reasons. 

79.  Are  all  circles  similar  figures?  Are  all  squares 
similar  figures?     Are  all  rectangles  similar  figures? 

Note.  —  Figures  of  the  same  shape  are  said  to  be  Similar. 

80.  Draw  a  rectangle  10  centimeters  long  and  4  centi- 
meters wide,  and  another  5  centimeters  long  and  2  centi- 
meters wide.  Are  their  homologous  sides  proportional, 
that  is,  is  the  ratio  of  their  widths  equal  to  the  ratio  of 
their  lengths  ?     Are  the  rectangles  similar  ? 

Note.  —  Similar  Polygons  are  those  whose  homologous  angles  are 
equal,  and  whose  homologous  sides  are  proportional. 

81.  Draw  a  rectangle  4  inches  by  3  inches,  and  another 
rectangle  12  inches  long  and  of  such  a  width  that  tlie 
homologous  sides  of  the  two  rectangles  shall  be  in  propor- 
tion, and  show  that  the  rectangles  are  similar. 

82.  Draw  a  rectangle  8  inches  by  6  inches,  and  another 
12  inches  by  7  inches.  Are  they  similar  ?  If  not,  change 
their  dimensions  so  as  to  make  them  similar. 

83.  Construct  two  similar  triangles  by  making  the 
homologous  angles  equal  and  the  including  sides  propor- 
tional. 

84.  Show  that  the  perimeter  of  a  rectangle,  20  milli- 
meters long  and  5  millimeters  wide,  has  the  same  ratio  to 
the  perimeter  of  one  80  millimeters  long  and  20  milli- 
meters wide,  as  any  two  homologous  sides. 


142 


POLYGONS. 


8 

1        « 

4 


85.  Given   the   rectangles ;    find   the   perimeters   and 
show  that  their  ratio  is  the  same  as 
that  of  the  longer  sides.     Is  it  the 
same  as  that  of  the  shorter  sides  ? 
Are  the  rectangles  similar? 

Principle  43.  —  TJie  perimeters  of  similar  polygons 
are  to  each  other  as  any  two  Ivontologoas  sid^s. 

86.  Given  the  similar  triangles  ABC  and  D£F;    sub- 
stitute the  numbers,  and  show  that  the 
following  proportions  are  true: 

AB:DE=BO:EF, 

AB:BF=AC:DF, 

BO:BF=:AO:BF, 

AB  +  BO-\-AC:BF+EF-\-BF 

=  AB:  DK 
Show  that  the  perimeters  of  the  trian- 
gles are  in  the  same  ratio  as  BC  and  EF;  also  in  the 
same  ratio  as  AC  and  DF. 

87.  All  similar  polygons  can  be  divided  by  diagonals 

into  an  equal  number  of  similar 
triangles  placed  similarly.  Re- 
produce the  given  pentagons 
by  constructing  triangles  of 
the  given  dimensions,  the  units 
of  length  being  millimeters  or 
centimeters.  Show  that  the  perimeters  of  the  pentagons 
are  proportional  to  any  pair  of  homologous  lines. 

88.  Draw  two  rectangles  whose  homologous  sides  are 
m  the  ratio  of  3  to  1,  and  show  that  tlieir  perimeters  are 
in  the  same  ratio.     Are   their  angles   equal?     Are  the 

rectangles  similar? 


POLYGONS. 


143 


89.  Construct  two  triangles  whose  homologous  sides 
are  in  the  ratio  of  2  to  3,  and  show  that  their  perimeters 
are  in  the  same  ratio. 

90.  The  perimeter  of  an  irregular  pentagon,  one  of 
whose  sides  is  6  inches,  is  33  inches.  Find  the  homologous 
side  of  a  similar  pentagon  whose  perimeter  is  60  inches. 

91.  Two  homologous  sides  of  two  similar  octagons 
are  respectively  5  centimeters  and  9  centimeters.  The 
perimeter  of  the  less  octagon  is  35  centimeters.  Find  the 
perimeter  of  the  greater. 

92.  The  sides  of  a  pentagon  are  5,  6,  7,  8,  and  9  inches. 
Find  the  perimeter  of  a  similar  pentagon  the  shortest  side 
of  which  is  10  inches. 

93.  The  longest  side  of  a  heptagon  is  10  inches,  and  its 
perimeter  is  24  inches.  Find  the  perimeter  of  a  similar 
heptagon  whose  longest  side  is  8  inches. 

94.  The  sum  of  the  bases  of  the  greater  of  two  similar 
isosceles  trapezoids  is  3  feet,  and  the  lower  base  is  twice 
the  upper.  Each  leg  is  J  of  the  upper  base.  The  greater 
base  of  the  less  trapezoid  is  8  inches.  How  many  inches 
are  there  in  the  perimeter  of  the  less  trapezoid  ? 

95.  AB  is  divided  into  four  equal  parts 
by  the  lines  drawn  parallel  to  BC.  Find 
four  triangles  whose  homologous  angles 
are  equal.  If  the  perimeter  of  the  triangle 
AJK  is  12  inches,  find  the  perimeter  of 
the  others. 

Note.  —  Triangles  which  are  mutually  equian- 
gular are  similar.  How  does  a  line  parallel  to  the 
base  of  a  triangle  divide  the  remaining  sides  V 


144 


POLYGONS. 


96.  If  the  triangle  DEF  is  superposed  upon  the  siraihir 
triangle  BAC^  can  the  angle  at  D 
be  made  to  coincide  with  the  angle 
at  ^?  Why?  BE  being  placed 
upon  the  homologous  side  BA^  will 
angle  BEFhe  equal  to  angle  BAC? 
Why?  Is  angle  BFE  equal  to 
angle  BOA?  U  tlie  angles  are 
equal,  will  the  line  FE  diverge 
from  BO  in  the  same  direction  as 
that  in  which  CA  diverges  from  BO,  or  will  it  l)e  parallel 
to  CA?  If  BA  is  10  niches,  BO  D  inches,  BE  5  inches, 
how  long  is  BF? 

97.  BE  is  parallel  to  BC.  Prove  by 
quoting  a  geometric  principk*  that  angle 
ABE=cmgle  ABC,  and  aiigk'  AED  =  angle 
A  CB.  Prove  by  quoting  that  AD :  DB  = 
AE :  A  C.  Substitute  numl^ers  for  the  terms 
of  this  proportion  and  show  that  ABiAD^ 
ACiAE. 


Principle  44.  —  J^  line  draiun  parallel  to  one  aids 
of  a  triangle  forms  with  tlie  other  two  sides  a  triangle 
similar  to  the  given  triangle. 

98.  i>^  is  parallel  to  ^(7. 
AB=^1  inclies,  AC=  10  inches,  BC=       j, 

11   inches,   DB  =  b   inches.     Find  DA,       a 
DE,  and  BE.  ^ 

99.  AB  and  DE  are  botli  perpen- 
dicular to  BC  BC=  10  fe^^t,  EC=  3 
feet.     DE=:4:  feet.     Find  AB. 


POLYGONS. 


145 


1      \ 

I           \ 

\ 

1 

i 

« 

\D 

? 

A„ 

100.  A  is  an  inaccessible  point  directly- 
above  B.  BO  is  20  feet.  BU  is  a  stick 
5  feet  high  placed  so  that  its  top  when 
sighted  from  0  is  in  a  line  with  A,  and 
its  base  is  4  feet  from  0.  Find  the 
height  of  A. 


101.  When  two  triangles  are  similar, 
can  the  smaller  always  be  applied  to  the  greater  in  such 
a  Avay  that  they  coincide  throughout  the  whole  extent  of 
the  smaller.  Why  ?  Prove  from  the  definition  of  similar 
triangles  that  the  homologous  sides  which  do  not  coin- 
cide are  parallel. 

102.  Triangle  ABCis  simi= 
lar  to  triangle  BEF. 

AB  =  ^  inches.  BE=^1 
inches.  i> J.  =  2^  inches.  AO 
=  4  inches.     BC=  5  inches. 

Find  OF.     Find  the  perimeter  of  each  triangle. 


103.  What  is  the  height  of  a  tree  whose  shadow  is 
40  feet  at  the  same  time  that  a  stick  5  feet  high  casts  a 
shadow  4  feet  in  length  ? 

104.  Mary  and  Anna  are  facing  towards  the  east. 
Mary,  who  is  4  feet  6  inches  tall,  casts  a  shadow  before  her 
2  feet  3  inches  long.  How  tall  is  Anna,  whose  shadow  is 
2  feet  6  inches  ?  In  which  part  of  the  day  can  the  condi- 
tions of  this  problem  actually  occur,  forenoon  or  after- 
noon ? 

105.  Define  all  the  terms  whose  meaning  has  been 
given  in  the  notes  of  this  chapter. 


HORN.    GEOM. 


10 


CHAPTER  XII. 

CIRCLES  AND  LINES. 

1.  How  many  circles  can  there  be  which  have  a  given 
point  as  a  center  ? 

Note.  —  Circles  having  the  same  center  are  Concentric.  '\Vhen 
they  are  equal  and  have  the  same  center,  they  are  Coincident. 

2.  Can  you  distinguish  by  the  eye  each  of  two  coinci- 
dent circles? 

3.  Draw  two  concentric  circles  and  name  the  figure 
which  is  included  between  their  circumferences. 

4.  Place  a  circle  whose  radius  is  6  inches  inside  a  cir- 
cle whose  radius  is  7  inches  so  that  their  circumferences 
touch,  and  find  the  distance  between  their  centers.  Find 
its  ratio  to  the  sum  of  the  radii.  To  the  difference  of 
the  radii. 

Note.  —  When  one  circle  is  placed  within  another  in  such  away 
that  they  are  not  concentric,  they  are  Eccentric,  and  if  their  circum- 
ferences touch  they  are  said  to  be  Internally  Tangent. 

5.  Can  you   draw  two  eccentric  circles  that   are  not 

tangent  internally  ? 

6.  Where  have  you  seen  wheels  and  rods  that  describe 
eccentric  circles  ?     Describe  their  working. 

T.  Draw  a  circle  and  place  within  it  several  other  cir- 
cles which  are  tangent  to  it,  making  the  inner  circles  equal 
to  one  another.    From  the  center  of  the  containing  circle, 

146 


CIRCLES  AND  LINES.  147 

with  a  radius  equal  to  the  distance  from  this  center  to  the 
center  of -one  of  the  inner  circles,  describe  a  circumfer- 
ence. Will  it  pass  through  the  centers  of  all  the  inner 
circles  ?      Defend  your  answer. 

8.  Draw  a  circle  with  a  radius  of  8  centimeters  and 
one  with  a  radius  of  7  centimeters,  making  them  externally 
tangent,  and  find  the  distance  between  their  centers. 
Compare  it  with  the  sum  of  their  radii. 

9.  Draw  two  circles  which  are  tangent  externally,  the 
radius  of  one  being  7  inches  and  of  the  other  5  inches. 
What  is  the  distance  between  their  centers?  What  is 
the  ratio  of  the  distance  between  their  centers  to  the  sum 
of  their  diameters  ?     To  the  sum  of  their  radii  ? 

10.  Draw  a  circle  and  several  other  circles  equal  to 
each  other,  externally  tangent  to  it.  Can  you  draw  a 
circle  whose  circumference  will  include  the  centers  of  all 
the  external  circles  ? 

11.  With  a  radius  of  3  inches  draw  a  circle.  With  a 
radius  of  1  inch  draw  five  circles  externally  tangent  to 
the  first.  Draw  a  line  from  the  center  of  the  inner 
circle  through  the  center  of  each  outer  circle  terminating 
in  its  circumference.  Find  the  length  of  each  of  these 
lines.  Draw  a  circumference  through  the  terminal  points 
of  the  lines  and  find  its  length.  Find  the  length  of  the 
circumference  which  passes  through  the  center  of  the 
outer  circles. 

12.  Where  have  you  seen  wheels  which  were  externally 
tangent  ?     Describe  their  working. 

13.  Draw  two  circles  which  intersect,  and  show  that  the 
distance  between  their  centers  is  less  than  the  sum  of 
their  radii. 


j^48  CIRCLES  AND  LINES. 

14.  Draw  one  circle  within  anotlier  and  not  tangent, 
and  show  which  is  greater,  the  distance  between  their 
centers  or  the  difference  of  their  radii. 

15.  Draw  two  circles  wholly  external,  and  show  which 
is  greater,  the  distance  between  their  centers  or  the  sum 
of  their  radii. 

16.  AB  is  a  tangent.  OC  is  the  radius  at  the  point 
of  tangency.  OE  and  OD  are  other  lines 
drawn  from  the  center  to  the  tangent.  F 
and  G-  are  the  points  where  they  cut  the 
circumference.  Which  is  the  greater,  OE 
or  OF'^  OF  or  OCi  Why  ?  OE  or  OCf 
OB  or  0Q'{  What  is  the  shortest  line 
that  can  be  drawn  from  a  point  to  a  line  ? 

Principle  45.  —  A  tangent  to  a  circle  iff  perperulic- 
ular  to  the  radius  at  tlie  point  of  tangency. 

17.  Referring  to  Ex.  16,  if  angle  DOC  is  40°,  how  many 
degrees  are  there  in  each  angle  of  the  triangle  COD'i 

18.  Cut  out  a  circle,  fold  it,  and  show  that  a  diameter 
bisects  the  circle  and  also  the  circumference. 

19.  CB  is  a  diameter  perpendicular  to  the  chord  AB  at 
the  point  E,     Reproduce  the  figure,  and  fold 

it  so  that  the  semicircle  CAB  is  superposed 
upon  the  semicircle  CBB.  Since  the  diame- 
ter bisects  the  circle,  will  the  Hue  AB  be 
bisected?  Will  the  arc  AB  equal  the  arc 
i)5?  Will  the  arc  AC  equal  the  arc  OB? 
Show  the  truth  of  your  answer. 

Principle  46. — A  radius  perpendicular  to  a  chard 
bisects  the  chord  and  the  arc  subtended  by  it. 


CIRCLES  AND  LINES.  149 

20.  AD^  4  centimeters,  is  perpendicular 
to  CB^  6  centimeters.  AE  is  5  centime- 
ters. A  is  the  center  of  the  circle.  Find 
the  sum  of  the  perimeters  of  the  triangles 
ABC  ?indi  ABB, 

21.  Draw  a  circle,  a  chord,  a  radius  perpendicular  to 
the  chord,  a  tangent  at  the  extremity  of  the  radius,  and 
radii  to  the  extremities  of  the  chord.  Prolong  the  radii 
until  they  meet  the  tangent,  and  show  that  four  similar 
triangles  are  formed  by  the  lines. 

22.  Referring  to  Ex.  20,  if  angle  GAB  is  80°,  how 
many  degrees  has  each  angle  of  the  triangle  ABC?  Of 
ABB'l     Quote  principles. 

23.  AB  is  perpendicular  to   CB  at  its  middle  point. 
If  the  distance  AC \^  10  inches,  how  long  ^ 
is   AB?     Quote   a    geometric    principle 
which  is  applicable. 

24.  Take  any  three   points   not   in  a 
straight  line,  draw  two  lines  connecting 

one  of  the  points  with  the  two  others,  and  at  the  middle 
points  of  those  lines  draw  perpendiculars.  Will  they 
meet  on  either  side  of  the  lines  ? 

25.  Draw  perpendiculars  at  the  middle  points  of  the 
lines  AB  and  BC,  which  form  an  angle.  From  the  point 
0,  where  the  perpendiculars  meet,  draw  lines  to  A^  B^ 
and  C.     Prove  OA,  OB,  and  OC  equal. 

26.  With  AO  as  a  radius,  draw  a  circle.  Will  the 
circumference  pass  through  B  and  C?  ^ 
Give  reason.                                                      ^ 

27.  Draw  a  circumference  through  the 
three  points  A,  B,  and  (7.  tf 


150  CIRCLES  AND  LINES, 

28.  Can  you  place  three  points  so  that  a  circumference 
cannot  be  made  to  pass  through  them  ? 

Principle  47.  —  Through  any  three  points  not  in  a 
straight  line  a  circumference  may  be  irvade  to  pass* 

29.  Take  three  points  not  in  a  straight  line  and  pass 
a  circumference  through  them.  Can  you  pass  more  than 
one  circumference  through  them  ?     Give  your  argument. 

30.  Take  two  points  and  see  how  many  circumferences 
can  be  made  to  pass  througli  them. 

31.  How  many  circumferences  may  be  drawn  through 

a  given  point? 

32.  Draw  an  isosceles  triangle  and  pass  a  circumfer- 
ence througli  the  vertices  of  its  angles.  Supply  the  miss- 
ing words  in  the  statements  —  The  circle  is .about 

the  triangle.     The  triangle  is in  the  circle. 

33.  How  many  isosceles  triangles  can  be  inscribed  in  a 
given  circle  if  the  equal  sides  of  all  the  triangles  meet  at 
a  given  point  ? 

34.  With  a  given  chord  as  a  base,  how  many  isosceles 
triangles  can  be  inscribed  in  a  given  circle  ? 

35.  Quote  the  geometric  principle  which  tells  how 
angles  at  the  center  of  a  circle  are  measured. 

36.  The  arc  BI)  is  3  times  the  arc  AB. 
Arc  AI)  =  1S0°.  How  many  degrees  are 
the  central  angles  A  OB  and  BOB? 

37.  Arc  AB  is  20°  less  than  arc  BC  in 
the  circle  whose  center  is  0.     Arc  ^C  is 
140°.     Find  the  number  of  degrees  in  an-  a 
gleAOB. 


CIRCLES  AND  LINES, 


151 


38.  Arc  5(7  is  31  times  arc  AC.  AB  is 
a  diameter,  and  0  is  its  middle  point.  How 
many  degrees  are  there  in  angle  AGO?  In 
angle  BOO? 

39.  Inscribe  an  angle  in  a  circle. 

Note.  —  An  angle  is  said  to  be  inscribed  in  a  circle  when  its  vertex 
is  in  the  circumference  and  its  sides  are  chords. 

40.  Angle  ABO  is  inscribed  in  the  circle  whose  center 
is  0.  The  arc  AC  is  60°.  We  wish  to  find 
the  number  of  degrees  in  the  inscribed 
angle.  Draw  the  construction  line  AO. 
How  many  degrees  are  there  in  the  exte- 
rior angle  A  00?  How  many  in  the  angles 
at  A  and  B  ? 

41.  Angle  ABO  is  inscribed  in  the  circle 
whose  center  is  0  The  arc  A  0=90°. 
How  many  degrees  are  there  in  the  angle 
ABO?  What  is  the  ratio  of  the  inscribed 
angle  ABO  to  the  central  angle  AGO? 

42.  Angle  ABO  is  inscribed  in  tlie  circle 
Avhose  center  is  G.  Arc  AB=oO°.  Arc 
UO=60°.  How  many  degrees  are  there  in 
the  angle  ABO? 

Suggestion.  —  Draw  the  diameter  BOE,  and  find 
the  number  of  degrees  in  each  angle  thus  formed. 

^     43.    If  there  are  50°  in  a  central  angle,  how  many  de- 
grees are  there  in  an  inscribed  angle  that  has  the  same  arc? 
44.    We  say  that ''  a  central  angle  is  measured  by  the  in- 
tercepted arc."     By  what  is  an  inscribed  angle  measured? 

Principle  48.  —  An  inscribed  angle  is  mea^sured  by 
one  half  the  arc  intercepted  by  its  sides. 


152 


CIRCLES  AND  LINES. 


45.  Arc  AB  is  8  times  the  arc  BC, 
J.  (7  is  a  diameter.  How  many  degrees 
are  there  in  the  angle  BAO?  How  long 
is  the  arc  BC,  ii  ACis  3 J  feet? 

46.  Arc  AB'iF.  60°  greater  than  arc  BC, 
Arc  BC  is  30°  greater  than  arc  CD.  AD 
is  a  diameter.  Ho\\'  many  degrees  are 
there  in  the  angle  BAC?  How  many  in 
the  angle  CAB? 

47.  AB  is  a  diameter.  How  many 
degrees  are  there  in  the  angle  ACB?  In 
the  angle  ^i)^? 


48.  The  three  chords  AB,  BC,  and  AC 
are  equal.  How  many  degrees  are  there 
in  each  angle  of  the  triangle  which  they 
form  ?     What  kind  of  a  triangle  is  it? 

49.  The  angle  BAC  is  S  times  the 
angle  CAB.  AB  is  a  diameter.  Arc 
-B^  =  80°.  How  many  degrees  are  there 
in  the  angle  BAC?  How  many  in  the 
angle  CAB? 


50.  The  vertical  angle  A  of  the  isosceles 
triangle  ABC  is  30°.  Find  the  number  of 
degrees  in  each  of  the  equal  arcs  AB  and 
AC.  How  long  is  each  arc  if  the  diame- 
ter of  the  circle  is  8  feet  2  inches  ? 


CIRCLES  AND  LINES. 


158 


51.  The  vertical  angle  A  of  the  isos- 
celes triangle  ABQ  is  3  times  a  base  angle. 
How  many  degrees  are  there  in  each  of  ^/ 
the  arcs  into  which  the  circumference  is 
divided  ?  If  angle  B  were  40°  less  than  an- 
gle A,  how  many  degrees  would  each  arc  be  ? 

52.  How  many  degrees  are  there  in  each  angle  inscribed 
in  a  semicircle  ? 

53.  The  arc  AC  i^  50°.  The  central 
angle  AEB  is  105°.  AD  is  a  diameter. 
How  many  degrees  are  there  in  each  an- 
gle of  the  triangle  AEB'}  How  many  in 
each  angle  of  the  triangle  DEC} 

54.  Arc  ^5  =  70°. 
Arc  i>(7=80°.     J.  (7  is  a  diameter,  and 

E  its  middle  point. 

How  many  degrees  are  there  in  each 
angle  of  the  triangles  AED  and  BE  CI 

55.  AB  and  CD  are  diameters. 
Arc  ^i>  =  48°. 
Find  each  angle  of  the  triangles  AOC 

and  DOB. 

Query.  —  Can  diameters  intersect  at  any  other 
point  than  the  center  of  the  circle  ? 

56.  Arc  2)5  =  130°.  Find  each  angle 
of  the  triangles  AOC  and  DOB,  DC  and 
AB  being  diameters. 

57.  Arc  DB,  which  equals  arc  AC,  is 
70°  greater  than  the  arc  AD  or  its  equal 
BC.  Find  each  angle  of  tlie  triangles 
AOC  and  DOB. 


154 


CIRCLES  AND  LINES. 


58.  Arc  BO  is  5  times  the  arc  AB. 
Find  all  the  angles  of  the  triangles  AOD 
and  BOC.O  being  the  intersection  of  the 
diameters  A  C  and  BB. 

59.  The  diameters  AD  and  BC  are  each 
10  inches.  The  chord  AB  is  8  inches. 
Find  the  perimeter  of  each  triangle. 
Quote  authority. 

60.  The  arc  DC  is  3°  greater  than  the 
arc  AD,  and  the  arc  AB  is  87®.  How 
many  degrees  are  each  of  the  angles  ABD 
and  1)^(7? 

61.  The  arc  ACB  is  108°.  Which  is 
the  greater  angle,  a,  ft,  or  <?  ? 

62.  Given  a  line  as  the  hypotenuse. 
Can  you  construct  a  right  triangle  ?  Can 
you  construct  more  than  one  right  triangle 
having  the  same  hypotenuse  ? 

63.  Are  all  the  right  triangles  which  can  be  constnicted 
on  a  given  hypotenuse  equal  in  area?     Argue  the  point. 

64.  How  long  is  the  radius  of  a  circle  which  is  cir- 
cumscribed about  a  right  triangle  whose  hypotenuse  is  15 

inches  ? 

Qb.  The  line  AB  drawn  across  a  circle 
cuts  the  circumference  into  two  arcs,  one 
of  which  is  140°  more  than  the  other. 
Find  each. 


Note.  —  A  line  which  cuts  a  circumference  in 
two  points  is  called  a  Secant. 


O 


CIRCLES   AND  LINES.  155 

QQ,  Inscribe  a  regular  hexagon  in  a  circle.  Bisect  each 
side  by  a  radius  of  the  circle.  Are  the  arcs  bisected? 
Quote  authority  for  your  answer.  Join  the  points  of 
bisection  of  the  arcs  to  the  extremities  of  the  chord.  Name 
the  polygon  formed  by  the  joining  lines.  If  the  arcs  sul> 
tended  by  the  sides  of  the  new  polygon  were  bisected  and 
the  points  of  bisection  joined  with  the  extremities  of  the 
sides,  a  polygon  of  how  many  sides  would  be  formed  ? 

67.  How  do  you  find  the  area  of  a  regular  polygon  ?  . 

68.  Draw  a  circle  and  inscribe  a  square  by  drawing  two 
diameters  at  right  angles  and  joining  their  extremities. 
Inscribe  a  regular  polygon  of  double  the  number  of  sides 
by  joining  the  extremities  of  each  side  of  the  polygon 
with  the  middle  point  of  the  arc  subtended  by  the  side. 
Double  the  number  of  sides  again,  and  so  on,  until  the 
sides  are  so  small  that  you  cannot  distinguish  the  perime- 
ter of  the  polygon  from  the  circumference  of  the  circle. 
How  would  you  find  the  area  of  the  last  polygon  that  you 
drew,  if  you  knew  its  side  and  apothem  ? 

69.  If  the  number  of  sides  of  a  regular  polygon  were 
indefinitely  increased  so  that  the  polygon  would  coincide^ 
with  the  circle,  to  what  would  the  perimeter  of  the  polygon 
be  equal  ?    What  would  the  apothem  of  the  polygon  equal  ? 

70.  Cut  a  circle  into  small  sectors.     Place  them  as  in 
the  figure.     The  sum  of  their 
bases  equals   what?     Their 
altitude  equals  what? 

71.  Considering  a  circle  as  a  polygon  of  an  infinite 
number  of  sides,  the  circumference  as  the  sum  of  those 
sides  and  the  radius  as  the  apothem,  can  you  see  the  truth 
of  the  following  theorem? 


l^Q  CIRCLES  AND  LINES. 

Principle  49.  —  The  area  of  a  circle  is  equal  to  one 
half  the  product  of  its  circumference  and  radius. 

72.  What  is  the  area  of  a  circle  whose  radius  is  14  cen- 
timeters? 

73.  What  is  the  area  of  a  circle  whose  diameter  is  21 
millimeters  ? 

74.  What  is  the  area  of  the  base  of  a  cone  whose  cir- 
cumference is  88  inches? 

75.  If  a  horse  is  tied  to  a  stake  on  a  lawn  by  a  rope  14 
feet  long,  over  how  many  square  feet  can  he  graze  ? 

76.  What  is  the  area  of  that  part  of  the  face  of  a  watch 
which  is  passed  over  in  an  hour  by  a  minute  hand  16 
millimeters  long  ? 

77.  What  is  the  area  of  a  sector  whose  angle  is  60®  of 
a  circle  Avhose  radius  is  7  inches? 

78.  What  is  the  area  of  a  sector  which  is  one  fourth  of 
a  circle,  if  its  arc  is  33  inches  long  ? 

79.  How  far  is  it  from  the  center  of  the  face  of  a 
nickel  to  its  edge?  Find  the  area  of  both  sides  of  a 
nickel  in  square  centimeters. 

80.  If  a  segment  is  cut  off  from  a  sector  which  has 
the  same  arc,  what  kind  of  a  figure  is  left  ? 

81.  Angle  A  OB  is  60°.     The  radius  A  0 
is  14  inches,  and  the  altitude  of  the  trian- 
gle AOB  is  12.12  +  inches.      Find  the  area  [        o<^    [)/, 
of  the  sector  A  OBD,  of  the  triangle  A  OB, 
and  of  the  segment  ABD. 

82.  A  square  is  inscribed  in  a  circle  whose  radius  is 
21  inches.  Find  the  area  of  each  of  the  segments  formed 
by  the  sides  of  the  square. 


CIRCLES  AND  LINES.  I57 

83.  A  square  is  circumscribed  about  a  circle  whose 
diameter  is  4  feet  8  inches.  Find  the  perimeter  of  each 
of  the  four  figures  bounded  by  the  arc  of  a  quadrant 
and  one  half  of  each  of  two  adjacent  sides  of  the  square. 
Find  the  area  of  each  figure. 

84.  A  square  whose  side  is  10  feet  is  placed  \vithm 
a  circle  whose  radius  is  24^  feet.  What  is  the  area  of 
the  surface  between  the  boundaries  of  the  square  and 
circle  ?  How  many  vertices  of  the  square  can  be  in  the 
circumference  ?     Illustrate. 

85.  A  circle  3J  inches  in  diameter  is  placed  within 
a  square  5  inches  in  dimensions.  Find  the  area  of  the 
surface  between  the  boundaries  of  the  figures. 

86.  A  rectangular  garden  20  rods  long  and  15  rods 
wide  has  a  circular  fountain  within  it  whose  circumfer- 
ence is  363  feet.  How  many  square  feet  of  space  are 
given  to  the  fountain,  and  how  much  remams  ? 

87.  The  diameter  of  the  larger  circle  is 
11^  inches,  the  diameter  of  the  smaller  is 
101  inches.  Find  the  area  of  the  circular 
ring  which  is  the  difference  between  them. 

88.  The  diameter  of  the  large  circle  is  14 
inches.  The  diameter  of  the  small  circle  is  8 
inches.  The  area  of  the  small  circle  is  wliat 
part  of  the  area  of  the  large  circle  ?  The  ring 
is  what  fractional  part  of  the  large  circle  ? 

89.  What  is  the  area  of  a  sector  of  120"*  of  a  circle 
whose  radius  is  8|  inches? 

90.  Find  the  area  of  a  sector  of  150**  of  a  circle  whose 

circumference  is  12^  centimetei-s. 


158  CIRCLES  AND  LINES. 

91.  How  many  square  inches  of  silk  are  used  for  both 
sides  of  a  fan  which  when  opened  is  a  semicircle,  the 
sticks  being  14  inches  long,  and  7  inches  of  the  sticks 
next  the  pivot  being  uncovered  ? 

92.  How  many  square  rods  are  there  in  the  area  of  a 
circular  race  track,  if  its  outer  edge  is  1  mile  in  circumfer- 
ence and  its  inner  edge  is  4400  feet  in  circumference  ? 

93.  How  many  square  inches  of  tin  are  required  to 
make  the  bottoms  of  a  dozen  pails,  the  diameter  of  each 
being  12  inches? 

94.  How  many  square  feet  are  tliere  in  the  area  of  two 
flower  beds,  each  1320  feet  in  circumference  ? 

95.  Find  the  areas  of  two  circles,  the  diameter  of  one 
being  3  times  that  of  the  other,  and  show  the  ratio  of 
their  areas. 

96.  Find  the  areas  of  two  circles,  the  diameter  of  one 
being  5  times  that  of  the  other,  and  show  the  truth  of 
the  following  principle: 

Principle  50.  —  The  areas  of  circles  are  to  ea^ch  other 
as  the  squares  of  tJieir  diameters,  or  as  tJie  sqimres  of 
their  radii. 

97.  How  many  times  is  the  area  of  a  circle  whose 
radius  is  2  inches  contained  in  the  area  of  a  circle  whose 
diameter  is  8  inches?  Solve  without  finding  the  area 
of  either  circle. 

98.  How  many  times  is  the  area  of  a  circle  whose 
radius  is  3  inches  contained  in  the  area  of  one  whose 
radius  is  7  inches? 


CIRCLES  AND  LINES. 


159 


99.  The  diameter  of  a  circle  whose  area  is  28.57  + 
square  inches  is  4  times  that  of  another  circle.  Find 
area  of  smaller  circle. 

Suggestion.  —  Let  x  =  required  area.    42:12  =  28.57  :  x. 

100.  The  radius  of  a  circle  whose  area  is  41.7  square 
inches  is  3  times  that  of  another  circle.  Find  the  area  of 
the  smaller  circle. 

101.  The  diameter  of  a  round  flower  bed  which  holds 
1250  plants  is  5  times  that  of  a  similar  one  in  which  the 
plants  are  similarly  placed.  How  many  plants  will  the 
smaller  bed  hold  ? 

102.  A  is  the  center  of  the  circle  in 
which  the  small  circles  are  placed.  Each 
small  circle  is  what  fractional  part  of  the 
large  circle  ?  The  irregular  figure  BCDA^ 
formed  by  the  upper  semi-circumferences 
of  the  three  circles  is  what  fractional  part 
of  the  large  circle  ? 

103.  AEB  and  BBQ  are   semicircum- 
ferences.     The  irregular  figure  AEBDCF  . 
is  what  part  of  the  semicircle  ACF  ? 

104.  AB  =  BC=CB.  The  semicircle 
X  is  what  iDart  of  the  semicircle  ABF? 
y  is  what  part  of  ABE  ?  z  is  what  part 
of  ABE  ?  The  irregular  figure  composed 
of  X  and  z  is  what  part  of  the  whole 
circle  ? 

105.  Find  the  area  of  all  the  surfaces  of  a  cylinder 
whose  bases  are  20  centimeters  in  diameter  and  whose 
altitude  is  11  centimeters. 


CHAPTER   XIII. 
CUMULATIVE    REVIEW   NO.  4. 

1.  When  are  two  lines  said  to  be  perpendicular  to  each 
other  ? 

2.  What  is  the  ratio  of  an  angle  of  20®  to  its  sup- 
plement ?     To  its  complement  ? 

3.  Find  the  exterior  angle  of  a  triangle  if  the  opposite 
interior  angles  are  20°  So'  and  70°  25'. 

4.  Is  an  isosceles  triangle  a  regular  polygon  ?     Defend 

your  answer. 

5.  Find  the  sum  of  the  perimeters  of  all  the  2-inch 
squares  into  which  an  8-inch  square  can  be  divided. 

6.  Find  the  ratio  of  the  sum  of  the  perimeters  of  all  the 
5-inch  squares  into  which  a  10-inch  square  can  be  divided 
to  the  perimeter  of  the  10-inch  square. 

7.  What  is  the  ratio  of  the  area  of  3  square  inches  to 
that  of  8  inches  square  ? 

8.  What  is  the  ratio  of  the  perimeter  of  the  square 
which  contains  4  square  inches  to  that  of  the  plane  figure 
which  is  4  inches  square  ? 

9.  Find  the  difference  between  the  area  of  a  rectangle 
containing  5  square  inches,  placed  side  by  side,  and  the 
area  of  a  plane  figure  which  is  5  inches  square. 

160 


CUMULATIVE  REVIEW.  l^\ 

10.  What  is  the  ratio  of  the  perimeter  of  the  figure 
which  contains  6  square  inches  placed  in  a  row  to  that 
of  the  figure  which  is  6  inches  square  ? 

11.  Find  the  side  of  a  square  whose  area  is  equal  to 
that  of  a  rectangle  16  inches  long  and  4  inches  wide. 
Find  the  ratio  of  the  perimeters  of  the  square  and  the 
rectangle. 

12.  The  side  of  a  square  is  18  inches.  One  of  the 
sides  of  a  rectangle  of  equal  area  is  54  inches.  Find  an 
adjacent  side.  Find  the  difference  between  the  perim- 
eters of  the  figures. 

13.  Find  the  perimeter  of  a  rectangle  whose  bases  are 
each  32  inches  and  whose  area  is  equal  to  that  of  a  square 
whose  side  is  24  inches. 

14.  Find  the  ratio  of  the  perimeter  of  a  square  whose 
area  is  64  inches  to  the  perimeter  of  a  rectangle  of  equal 
area  a  side  of  which  is  32  inches. 

15.  Find  the  difference  in  the  cost  of  fencing  a  square 
lot  whose  side  is  12  rods  and  a  rectangular  lot  of  equal 
area  one  of  whose  sides  is  36  rods,  the  fence  costing  10 
cents  a  foot. 

16.  Find  the  difference  in  the  cost  of  the  outside  foun- 
dation walls  of  a  house  which  contains  four  rooms,  each 
16  feet  square,  placed  in  a  row,  and  a  square  house  con- 
taining four  rooms  each  16  feet  square,  if  eiich  lineal  foot 
of  the  foundation  costs  50  cents  and  no  allowance  is  made 
for  partitions  or  corners. 

17.  What  will  be  the  difference  in  the  cost  of  the  floor- 
ing of  the  houses  if  each  square  foot  costs  18|  cents  ? 

HORN.    GEOM.  —  11 


162  CUMULATIVE  REVIEW. 

18.  Given  a  square,  and  a  rectangle  of  equal  area 
which  is  not  a  square.  Which  has  the  greater  perim- 
eter ?     Argue  the  case. 

19.  Of  rectangles  having  equal  perimeters,  which  has 
the  greater  area,  a  square  or  a  rectangle  which  is  not  a 

square  ?     Illustrate. 

20.  Draw  a  rhomboid  and  a  rectangle  having  equal 
bases  and  perimeters  and  show  which  has  the  greater 
area. 

21.  Find  the  difference  in  area  between  a  square  whose 
boundary  is  88  inches  and  a  circle  whoso  boundary  is  88 
inches. 

22.  When 'Queen  Dido,  landing  on  the  shores  of  Africa, 
wished  to  inclose  the  greatest  amount  of  land  by  the  strips 
of  hide,  should  she  have  laid  them  so  as  to  inclose  a  square 

or  a  circle  ? 

Suggestion.  —  See  encyclopedia  or  classical  dictiouary  for  the 
story  of  Queen  Dido.  ' 

23.  The  line  AB  cuts  the  circumference 
into  two  arcs,  one  of  which  is  4  times  the 
other.  How  many  degrees  are  there  in  each 
arc?  What  name  is  given  to  a  line  which 
cuts  a  circumference  in  two  points? 

24.  Draw  a  circle  and  two  secants  meeting  at  a  point 

without  the  circle. 

25.  Draw  a  circle,  a  chord,  and  radii  to  the  extremities 
of  the  chord.  Join  the  middle  point  of  the  chord  with 
the  center  of  the  circle.  What  kind  of  triangles  are  thus 
formed  ?     Quote  geometric  principle. 


CUMULATIVE  REVIEW.  163 

26.  AC \&  parallel  and  equal  to  BD. 
Arc  CD  =  60°.     Find  all  the  angles  of  the 

triangles  AOC  and  BOD. 

27.  0  is  the  middle  point  of  the  diameters 
AD  and  BC.  Arc  BD  equals  4  times  the 
arc  AB. 

Find  each  angle  of  the  triangles  COA 
and  BOD. 

28.  Show  how  you  circumscribe  a  circle  about  a  scalene 
triangle. 

29.  Given  a  circle.  Inscribe  a  scalene  triangle.  An 
isosceles  triangle. 

30.  In  the  triangle  ABC  angle  A  —  two  times  angle  B 
and  angle  0  is  33°.  How  many  degrees  are  there  in 
angle  A  and  in  angle  B  ? 

31.  Find  the  sum  of  the  exterior  angles  made  by  pro- 
longing the  equal  sides  of  an  isosceles  triangle  whose  ver- 
tical angle  is  45°. 

32.  Can  you  draw  an  isosceles  right  triangle?  Can 
you  draw  an  isosceles  triangle  having  its  base  angles  right 
angles?     Can  you  draw  an  equilateral  right  triangle? 

33.  Can  you  divide  an  equilateral  triangle  into  equal 
triangles  by  one  line  ?  An  isosceles  triangle  ?  A  scalene 
triangle  ? 

34.  Can  you  divide  a  scalene  triangle  into  two  triangles 
which  are  equivalent  ? 

Note.  —  Observe  the  distinction  between  equal  and  equivalent. 
Equal  polygons  are  those  which  can  be  made  to  coincide  in  all  their 
parts.    Equivalent  polygons  are  those  which  have  the  same  area. 


154  CUMULATIVE  REVIEW. 

35.  Can  a  square  be  equal  to  a  rhomboid?  Can  it  be 
equivalent  to  a  rhomboid? 

36.  ABCD  is  a  rhomboid. 

AB  =  9  inches.      EF,  the  altitude,  =  4  c   b  n 

inches.      Find    the   side   of  a   square   of        /  / 

equal   area.      Wliat   is   the    ratio   of   the    ^        f      b 
perimeters  of  the  figures? 

37.  The  area  of  a  rhomboid  is  240  square  inches  and 
one  of  the  sides  is  2  feet.  Find  the  altitude  perpendicular 
to  that  side. 

88.  The  area  of  a  rhomboid  is  270  square  feet  and  one 
of  the  altitudes  is  10  feet.     Find  the  corresponding  base. 

39.  The  side  AD  of  the  trapezium  ABCD  is  8  inches 
longer  than  the  side  DC.  DC  is  2  inches  longer  than  BC^ 
BC equals  AB,  and  the  perimeter  is  27  inches.  Vind  each 
side. 

40.  Can  you  draw  a  trapezoid  liaving  a  right  angle  at 
each  extremity  of  one  of  the  non-parallel  sides? 

41.  Can  you  draw  a  trapezoid  having  a  right  angle  at 
each  extremity  of  a  base  ? 

42.  Find  the  angles  of  a  rhombus  each  of  whose  obtuse 
angles  is  three  times  a  consecutive  angle. 

43.  Find  the  perimeter  of  the  regular  hexagon  inscribed 
in  a  circle  whose  radius  is  9  centimeters. 

44.  A  regular  hexagon  is  inscribed  in  a  circle  whose 
radius  is  31^  centimeters.  Find  the  sum  of  the  perimeters 
of  the  three  equal  rhombuses  into  which  the  hexagon  can 
be  divided  by  three  radii. 


CUMULATIVE  REVIEW.  1^5 

45.  Inscribe  a  regular  hexagon  in  a  circle  and  join  the 
alternate  vertices.  What  figure  is  formed?  What  is  its 
ratio  to  the  hexagon? 

46.  Show  how  you  inscribe  an  equilateral  triangle  in  a 

circle. 

47.  Can  you  divide  a  regular  hexagon  into  twelve  equal 
scalene  triangles  ? 

48.  Can  you  divide  a  regular  hexagon  into  six  equal 
isosceles  triangles? 

49.  Can  you  divide  a  regular  hexagon  into  six  equal 
equilateral  triangles  ? 

50.  Inscribe  a  hexagon  in  a  circle.  Draw  radii  to  the 
vertices  of  the  hexagon,  draw  tangents  perpendicular  to 
those  radii,  and  prolong  them  until  they 
meet.  You  have  circumscribed  a  hexagon 
about  a  circle.'  Fold  the  outer  triangles, 
as  the  triangle  ABO^  upon  the  hexagon 
and  discover  what  part  an  inscribed  hex- 
agon is  of  a  circumscribed  hexagon. 

Query. —  How  do  the  triangles  ADO,  BOD,  and  ABD  compare, 
D  being  the  middle  point  of  the  triangle  0AB1 

51.  If  any  point  on  a  circumference  is  joined  with 
the  extremities  of  a  diameter,  what  kind  of  a  triangle  is 
formed  ? 

52.  Find  the  ratio  of  the  areas  of  two  circles,  the 
diameter  of  one  being  1  decimeter  and  of  the  other  15 
centimeters. 

53.  Find  the  ratio  of  the  areas  of  two  circles,  the 
diameter  of  one  being  5  meters  and  the  radius  of  the 
other  being  5  decimeters. 


166  CUMULATIVE  REVIEW. 

64.  Find  the  area  of  an  isosceles  triangle  whose  base 
is  5  decimeters  and  altitude  10  centimeters.  Find  the 
area  of  a  scalene  triangle  having  the  same  base  and  alti- 
tude, and  compare  the  areas  of  the  triangles. 

55.  Reproduce  the  right-angled  scalene  tri- 
angle ABC^  whose  base  is  4  centimeters  and 
altitude  7  centimeters,  and  construct  an  isosce- 
les triangle  having  the  same  base  and  area. 

56.  Find  the  fourth  proportional  to  three  lines  respec- 
tively 6  inches,  8  inches,  and  12  inches. 

57.  What  line  has  the  same  ratio  to  a  line  16  inches 
long  that  a  line  5  inches  long  has  to  one  8  inches  long? 

58.  How  long  is  a  line  whose  ratio  to  a  line  20  inches 
long  is  the  same  as  that  of  a  line  7  inches  long  to  a  line 
28  inches  long? 

59.  Find  the  ratio  of  the  area  of  a  rectangle  9  inches 
long,  5  inches  wide  to  one  10  inches  long,  3  inches  wide. 

60.  A  rectangle  9  inches  long,  7  inches  wide  is  in  the 
same  ratio  to  one  7  inches  long,  4  inclies  wide  as  a  square 
9  inches  in  dimensions  is  to  another  square.  Find  the 
perimeter  of  the  smaller  square. 

61.  What  is  the  ratio  of  one  of  the  angles  of  a  regular 
decagon  to  the  sum  of  its  exterior  angles  ? 

62.  A  semicircle  is  divided  into  two  sectors,  the  arc 
of  one  of  which  is  30°  more  than  that  of  the  other.  The 
greater  sector  is  what  part  of  the  semicircle  ? 

63.  Which  is  nearer  the  center  of  the  circle,  a  chord 
8  centimeters  long  or  one  6  centimeters  long  in  the  same 
circle  ?    Illustrate  and  quote  geometric  principle. 


CUMULATIVE  REVIEW.  167 

64.    Show  how  you  inscribe  a  square  in  a  circle. 

Qb.  Draw  a  circle  and  two  diameters  at  right  angles. 
At  each  extremity  of  each  diameter  draw  a  parallel  to  the 
other  diameter,  prolonging  them  until  a  surface  is  in- 
closed. What  names  belong  to  the  polygon  thus  formed 
by  the  four  tangents  ? 

QQ.  What  is  the  area  of  the  square  circumscribed  about 
a  circle  whose  radius  is  12  inches? 

67.  What  is  the  ratio  of  a  square  circumscribed  about 
a  circle  to  a  square  inscribed  in  the  same  circle  ? 

68.  The  end  of  a  round  log  is  12  inches  in  diameter. 
What  is  the  area  of  the  end  of  the  largest  square  stick 
of  timber  that  can  be  cut  from  it  ? 

69.  How  many  times  will  a  wheel  whose  circumference 
is  25  inches  revolve  in  traveling  30  feet  ? 

70.  How  many  times  will  a  wheel  whose  diameter  is 
63  inches  revolve  in  traveling  115  feet  6  inches  ? 

71.  The  base  of  an  isosceles  triangle  whose  perimeter 
is  80  inches  is  20  inches  less  than  twice  one  of  tlie  equal 
sides.  Find  the  length  of  the  greatest  common  measure 
of  the  sides  of  the  triangle.  How  many  times  can  it  be 
laid  off  on  the  perimeter  ? 

72.  Bisect  two  angles  of  an  equilateral  triangle,  and 
find  how  many  degrees  there  are  in  the  angles  formed  by 
the  bisectors.  How  many  degrees  are  in  the  angles 
formed  by  the  bisectors  of  the  base  angles  of  an  isos- 
celes triangle  whose  base  angles  are  each  twice  the  verti- 
cal angle  ? 

73.  The  perimeter  of  an  isosceles  triangle  whose  base 
is  its  shortest  side  is  100  centimeters.      The  difference 


168  CUMULATIVE  REVIEW, 

between  the  base  and  an  adjacent  side  is  23  centimeters. 
The  altitude  is  40  centimeters.     Find  the  area. 


74.  The  difference  of  angle  a  and  angle  h  p  ^7 
is  12°.     Find  each  angle  of  the  rhomboid.           \h ^ 

75.  The  longest  side  of  a  trapezium  whose  perimeter 
is  49  inches  is  21  inches.  Find  the  longest  side  of  a 
similar  trapezium  whose  perimeter  is  35  inches,  and  quote 
the  geometric  principle  upon  which  your  work  deixjnds. 

76.  Include  an  angle  of  120°  by  two  lines  each  5  centi- 
meters. At  the  end  of  one  of  the.  luies  make  on  the 
same  side  another  angle  of  120°  with  a  line  5  centimeters. 
Continue  this  operation  until  you  have  inclosed  a  surface, 
and  write  its  name  upon  it. 

77.  Take  Ex.  76,  substituting  135°  for  120**. 

78.  Construct  a  regular  hexagon  whose  sides  are  each 
5  centimeters.  Its  apothem  is  4.33+  centimeters.  Wliat 
is  its  area  ? 

Note.  — If  the  hexagon  were  to  rise  in  the  air,  keeping  itself 
always  parallel  to  its  first  position,  the  geometric  solid  which  its 
path  would  form  would  be  a  Hexagonal  Prism. 

79.  If  the  hexagon  in  Ex.  78  rose  to  the 
height  of  10  inches,  what  would  be  tlie  area 
of  all  the  surfaces  of  the  hexagonal  prism  ? 

80.  What  is  a  prism  ?     What  are  its  bases  ? 
Suggestion.  —  See  dictionary. 

81.  Find  the  lateral  surface  of  a  triangular 
prism,  each  side  of  the  bases  being  3  feet  and  the 
height  of  the  prism  being  9  feet. 


CUMULATIVE  REVIEW.  169 

82.  Find  all  the  surfaces  of  a  regular  pentagonal  prism, 
one  side  of  a  base  being  8  feet,  the  apothem  being  5.49-h 
feet,  and  the  altitude  of  the  prism  12  feet. 

83.  What  is  the  height  of  a  flag  pole  whose  shadow  is 
5  feet  when  the  shadow  of  an  upright  stick  4  feet  long 
is  3  inches? 

84.  The  length  of  a  stick  and  the  length  of  its  shadow 
when  placed  upright  are  equal.  At  the  same  time  what 
is  the  length  of  the  shadow  of  a  steeple  48  feet  high  ? 

85.  The  line  AD^  which  shows  the  alti- 
tude of  the  triangle  ABC^  when  BC  is  con- 
sidered the  base,  is  4  inches.  BO  is  Q  inches, 
BU,  the  altitude,  when  AC  is  considered  the 
base,  is  3  inches.     Find  A  0. 

86.  Arrange  the  following  in  three  lists,  one  of  lines, 
one  of  surfaces,  the  other  of  solids  :  Chord,  circle,  liexa- 
gon,  octagon,  cube,  segment,  arc,  quadrilateral,  sector, 
trapezium,  pyramid,  polygon,  cone,  perimeter,  diameter, 
sphere,  semicircle,  circumference,  radius,  rectangle,  quad- 
rant, prism,  liter,  triangle,  measure  of  an  arc,  measure  of 
a  sector,  cylinder,  se'micircumference,  decagon,  rhombus, 
measure  of  a  line,  trapezoid,  plane,  diagonal,  rhomboid, 
transversal,  dodecagon,  secant,  median,  square,  parallelo- 
gram, tangent,  parallelopiped. 


CHAPTER  XTV. 

SQUARES  AND  CUBES. 

1.  Complete  the  table  of  the  squares  of  the  integers 
from  1  to  25  inclusive.     1*  =  1,  2^  =  4,  etc. 

Make  a  table  of  the  squares  of  numbers  expressed  by 
a  significant  figure  and  a  cipher,  as  10,  20,  30,  and  show 
how  the  second  table  is  derived  from  the  first. 

2.  Find  with  the  help  of  your  tables  the  side  of  a 
square  whose  area  is  484  square  meters.  Of  one  whose 
area  is  48,400  square  meters. 

3.  Find  the  side  of  a  square  whose  area  is  529  square 
feet.     Of  one  whose  area  is  52,900  square  feet. 

4.  Find  the  length,  width,  and  area  of  the  rectangle 
formed  by  placing  two  squares,  each  containing  361  square 
inches,  so  that  they  have  a  common  side.     Illustrate. 

5.  Find  the  length  of  a  rectangle  twice  as  long  as 
broad  whose  area  is  1152  square  inches. 

6.  Find  the  perimeter  of  a  square  whose  area  is  441 
square  meters. 

7.  Find  the  perimeter  of  a  rectangle  3  times  aa  long 
as  broad  whose  area  is  588  square  meters. 

8.  What  line  squared  and  multiplied  by  3  gives  a 
rectangle  containing  768  square  inches  ? 

9.  If  X  represents  the  side  of  a  square,  what  repre- 
sents its  area?  The  rectangle  which  contains  three  such 
squares  ?     The  width  of  the  rectangle  ?     Its  length  ? 

170 


SQUARES  AND  CUBES,  17J 

10.  Find  the  width  and  length  of  a  rectangle  wliich  is 
5  times  as  long  as  wide,  and  contains  2205  square  inches. 

11.  A  rectangular  lot  whose  length  is  4  times  its  width 
contains  676  square  rods.     Find  its  dimensions. 

12.  How  many  yards  of  binding  will  it  take  for  a 
square  oilcloth  mat  covering  81  square  feet  ? 

13.  Of  two  squares,  the  greater  contains  24  square  feet 
more  than  the  less.  The  sum  of  their  areas  is  74  square 
feet.     Find  the  side  of  each. 

Suggestion.  —  Let    x  =  side  of  smaller  square. 

Then  x^  =  area  of  smaller  square, 

x^  +  24  =  area  of  greater  square. 

14.  The  first  of  four  squares  contains  63  square  inches 
more  than  the  second,  the  second  17  square  inches  more  than 
the  third,  the  third  28  square  inches  more  than  the  fourth. 
Their  total  area  is  325  square  inches.     Find  tlie  side  of  each. 

15.  There  are  three  squares,  the  first  of  which  is  4 
times  the  second,  and  the  second  is  9  times  the  third. 
Their  combined  area  is  414  square  feet.  Find  the  length 
of  their  combined  perimeters. 

16.  Find  the  number  of  rods  of  fencing  required  to 
inclose  separately  three  square  lots,  the  first  of  which 
contains  11  square  rods  more  than  the  second,  and  the 
second  contains  16  square  rods  more  than  the  tliird,  their 
combined  area  being  70  square  rods. 

17.  Three  squares  are  arranged 
as  in  the  diagram.  They  cover 
184  square  inches. 

Find  the  length  of  the  boundary 
line  of  the  surface  covered  by  them 
if  the  square  on  the  left  is  4  times  as  great  as  the  middle 


172  SQUARES  AND  CUBES. 

square,  and  the  middle  square  is  9  times  as  great  as  the 
one  on  the  right. 

18.  The  perimeter  of  a  certain  square  is  4  inches  longer 
tlian  that  of  another  square,  and  the  sum  of  their  perime- 
ters is  100  inches.     Find  the  sum  of  their  squares. 

19.  How  many  times  is  the  square  of  any  number  con- 
tained in  the  square  of  twice  that  number  ?     Illustrate. 

20.  Draw  squares,  and  show  how  many  times  the  square 
of  a  4-inch  line  is  contained  in  the  square  of  an  8-inch. 
In  the  square  of  a  12-inch  line. 

21.  How  many  times  is  the  square  of  any  number  con- 
tained in  the  square  of  3  times  that  number  ? 

22.  How  many  times  does  the  square  of  a  line  contain 
the  square  of  ^  of  that  line  ? 

23.  A  square  inch  can  be  divided  into  how  many  figures 
each  J  of  an  inch  square  ? 

24.  Draw  a  figure  which  is  tlie  sum  of  the  squares  of 
two  lines,  one  8  inches,  the  other  3  inches,  and  find  its 
area. 

25.  Draw  a  figure  which  is  the  square  of  the  sum  of 
the  same  lines,  and  find  its  area. 

26.  Find  the  difference  between  the  sum  of  the  squares 
and  the  square  of  the  sum  of  two  lines,  one  7  inches,  the 
other  8  inches. 

27.  Draw  a  rectangle  which  is  the  product  of  two  lines 
respectively  8  inches  and  5  inches,  and  find  its  area  and 
perimeter. 

28.  Let  AB  and  BC  be  two  lines  respectively  6  inches 
and  4  inches,  and  AC  their  siun. 


SQUARES  AND  CUBES. 


173 


G             H         IT 

E 

F 

A 

B 

C 

How  many  square  inches  are  there  in  the 
square  ADUB? 

How  many  square  inches  are  there  in  the 
square  jEHKF? 

How  many  square  inches  are  there  in  the 
rectangle  BUFC? 

How  many  square  inches  are  there  in  the  rectangle 

Daffu? 

How  many  in  all  ? 

Principle  51.  —  The  square  of  tJie  sum  of  two  lines 
is  equal  to  the  square  of  the  first  plus  twice  tlie  prod- 
uct of  the  first  hy  the  second  plus  the  square  of  the 
second. 

29.  Draw  the  square  of  the  sum  of  two  lines,  one  5 
inches,  the  other  3  inches,  and  show  the  truth  of  Prin.  51. 

30.  Square  the  sum  of  a  and  h,  a  representing  a  line  7 
inches,  and  h  representing  a  line  2  inches. 

31.  DraAv  a  figure,  and  show  that  (a  +  ^)^  =  a*  -|- 
2(a  X  i)  +  62  when  a  =  a  10-inch  line  and  ^  =  a  3-inch 
line. 

32.  Let  a  and  h  be  two  lines.  Show  by 
numbers  that  the  square  of  their  sum  is 
represented  by  the  diagram. 

33.  a;  =  10  inches,  «/  =  a  line  less  than  10  inches,  and 
the  square  of  their  sum  =  289  square  inches.     Find  y. 

34.  a;  =  10,  «/  =  a  less  nmnber,  the  square  of  their  sum 
=  32-4.     Find  y. 

35.  a;  =  10    inches.      Find    by    trial    the 
value  of  y,  if  (a;-f  i/)2=169. 

36.  The  square  of  the  sum  of  two  lines  is 
256  square  inches;  the  greater  is  10  inches.    Find  tlie  less. 


ax  h 

h* 

o« 

3 

X         y 


174  SQUARES  AND   CUBES. 

37.  The  square  of  the  sum  of  two  lines  is  441  square 
inches;  the  greater  is  20  inches.     Find  the  less. 

38.  Find  the  sum  of  the  separate  perimeters  of  the 
squares  and  rectangles  which  make  the  square  of  the  sum 
of  a  line  20  inches  and  a  line  3  inches. 

39.  If  you  had  196  square  feet  of  lumber  which  you 
wished  to  place  in  the  form  of  a  square,  and  were  to  begin 
by  making  a  square  10  feet  each  way,  how  wide  must  the 
additions  be  in  order  to  complete  the  square  ? 

40.  Find  the  square  root  of  1849,  and  show  how 
Principle  51  assists  you  in  the  work.  Why  do  you 
double  the  first  figure  found  in  the  root  in  order  to 
obtain  a  trial  divisor? 

Suggestion.  —  See  arithmetic  for  rule  for  extracting  the  square 
root  of  numbers. 

41.  Find  the  square  root  of  1225,  2601,  5184,  3844, 
5329,  5929,  8649,  12769,  14884,  94249,  495616. 

42.  Lay  off  the  line  AB  4  inches  on  the  line  AD  7 
inches;  draw  a  square  on  the  difference,  and  find  its  area. 

43.  Construct  a  square  on  the  difference  of  the  lines 
11  inches  and  6  inches,  and  find  its  area.     Find  the  sum 

of  the  squares  of  those  lines. 

44.  Given  a  line  7  inches  and  one   3     4  ^'»-  ^     F 


inches.      How    many    square    inches    are    gRT j--  . 

there  in  the  square  of  their  difference  ?  ?:::::::" 
In  the  sum  of  their  squares?  How  many  Him. i: sin. l 
square  inches  are  there  in  the  rectangle  ABEQ-  ?  Which 
is  greater,  the  rectangle  ABEQ-  or  the  rectangle  KLFBl 
Reproduce  the  figure  which  is  the  sum  of  the  squares 
of  the  lines,  cut  out  the  square  of  their  difference  and  see 
what  remains. 


SQUARES  AND   CUBES.  175 

Principle  52.  —  The  square  of  the  difference  of  two 
lines  is  equal  to  the  sum  of  their  squares  minus  twice 
the  rectangle  of  the  lines. 

45.  Given  the  lines  x,  8  inches  and  y,  3  inches.  Cut 
out  the  figure  made  of  the  sum  of  the  squares  of  the  two 
lines,  and  show  by  superposing  that  if  the  square  of  the 
difference  of  these  lines  is  cut  from  the  figure,  the  re- 
maining space  will  be  equal  to  two  rectangles  of  these 
lines. 

Show  the  same  truth  about  other  lines  until  you  realize 
the  meaning  of  Prin.  52. 

46.  Draw  a  figure  and  show  the  truth  of  the  statement 
(a  — 5)2=^2  — 2 (ax J) +  62,  when  a  =  a  line  5  inches  and 
Z>  =  a  line  3  inches. 

47.  Draw  a  rectangle  which  is  the  product  of  the  sum 
of  two  lines  by  the  greater  line,  the  lines  being  8  inches 
and  5  inches,  and  find  its  perimeter. 

48.  Draw  a  rectangle  which  is  the  product  of  the 
difference  of  two  lines  by  the  greater,  and  find  its  area; 
the  lines  being  12  inches  and  9  inches. 

49.  There  are  two  lines  6  inches  arid  5  inches  respec- 
tively. Draw  the  rectangle  of  the  sum  of  the  lines  by  the 
less,  and  find  the  area  and  perimeter. 

50.  Given  one  line  5  inches,  another  3  inches.  Draw 
the  rectangle  of  their  sum  and  difference,  and  give  its 
area. 

51.  Draw  a  rectangle  which  is  the  product  of  the  sum 
and  the  difference  of  two  lines,  one  of  which  is  7  inches 
and  the  other  3  inches,  and  show  that  the  number  which 
represents  its  area  is  equal  to  72—32. 


176  SQUARES  AND   CUBES. 

Principle  53.  —  The  rectangle  of  the  sutjv  and  dip 
ference  of  two  lines  is  equal  to  the  difference  of  the 
squares  of  the  lines. 

52.  Draw  the  rectangle  which  is  the  product  of  the  sum 
and  difference  of  two  lines  respectively  7  inches  and  2 
inches.  Draw  the  figure  which  is  the  difference  of  the 
squares  of  those  lines,  the  smaller  square  being  subtracted 
from  the  upper  right-hand  corner  of  the  greater.  Super- 
pose this  figure  upon  the  first,  cutting  off  the  rectangle 
at  the  top,  which  is  the  product  of  the  less  line  by  the 
difference  of  the  lines,  and  applying  it  at  the  side.  Show 
the  application  of  Prin.  53. 

53.  Find  the  area  and  perimeter  of  the  rectangle  of  the 
sum  and  difference  of  two  lines  whose  sum  is  15  inches, 
one  of  which  is  twice  the  other. 

54.  Find  the  area  of  the  rectangle  of  the  sum  and 
difference  of  two  lines  whose  sum  is  15  inches,  one  line 
being  3  inches  longer  than  the  other,  and  show  by  diagrams 
how  much  this  rectangle  lacks  of  being  equal  to  the  square 
of  the  greater  line,  and  how  much  it  exceeds  the  square  of 
the  less. 

55.  Draw  a  right  triangle 
having  a  base  5  inches  and  per- 
pendicular 12  inches.  It  will 
be  found  that  the  hypotenuse 
is  13  inches.  Construct  a  square 
upon  each  side,  and  cut  out  the 
figure.  Apply  the  square  of  the 
perpendicular  to  the  square  of 
the  hypotenuse.  Cut  and  fit  the  square  of  the  base  to 
the  remaining  space. 


ti 

\ 

u 

6 

SQUARES  AND   CUBES.  177 

% 

56.  Draw  a  right  triangle  having  a  base  3  inches  and 
perpendicular   4    inches.       The   hypotenuse 
will  be  5  inches.     Construct  a  square  upon 
each  side,  and  apply  the  squares  of  the  base 
and  perpendicular  to  that  of  the  hypotenuse. 

Note.  — Demonstrative  geometry  will  show  the  truth  in  all  cases 
of  the  following  principle : 

PRINCIPLE  54.  —  The  square  of  the  hypotenuse  of  a 
right-angled  triangle  is  equal  to  the  sum  of  the  squares 
of  the  other  two  sides. 

57.  Measure  off  6  inches  on  one  side  of  a  square  sur- 
face and  8  inches  on  the  adjacent  side.  How  long  is  the 
line  that  joins  the  extremities  of  these  two  lines  ? 

58.  A  travels  30  miles  east  from  a  certain  point;  B  40 
miles  north  from  the  same  point.     How  far  apart  are  they  ? 

59.  Draw  a  rectangle  12  centimeters  long  and  9  centi- 
meters wide,  and  find  its  diagonal. 

60.  What  is  the  distance  from  the  lower  left-hand 
corner  to  the  upper  right-hand  corner  of  a  gate  which  is 
8  feet  long  and  6  feet  wide  ? 

61.  What  is  the  length  of  the  longest  stick  pointed  at 
both  ends  that  can  lie  wholly  on  a  table  which  is  24  feet 
long  and  7  feet  wide  ? 

Query.  —  Why  must  it  be  pointed? 

62.  What  is  the  distance  diagonally  across  a  rug  16 
feet  long  and  12  feet  wide  ? 

63.  A  vacant  lot  is  24  rods  long  and  18  rods  wide. 
Mary  crossed  it  diagonally,  and  Anna  reached  the  same 
point  by  walking  along  the  edge  of  the  lot.  How  much 
farther  did  Anna  walk  than  j\Iary  ? 

HORX.   GEOM.  —  12 


178 


SQUARES  AND  CUBES. 


64.  The  base  of  an  isosceles  triangle  is  30  feet,  and  it« 
altitude  is  36  feet.     Find  one  of  the  equal  sides. 

Query.  —  Into  what  two  equal  figures  is  an  isosceles  triangle 
divided  by  a  perpendicular  drawn  from  the  vertex  to  the  base  ? 

65.  The  base  of  an  isosceles  triangle  is  64  inches,  and 
its  altitude  is  24  inches.     Find  its  perimeter. 

^Q,  The  base  of  an  isosceles  triangle  is  6  inches,  and 
its  area  is  12  square  inches.  Find  its  altitude  and  one  of 
the  equal  sides. 

67.  Find  the  perimeter  of  an  isosceles  triangle  whose 
base  is  20  feet,  and  whose  altitude  is  24  feet. 

68.  The  base  of  a  right  triangle  is  15  centimeters,  and 
the  perpendicular  20  centimeters.  Their  sum  is  how 
much  more  than  the  hypotenuse  ? 

69.  Find  the  approximate  length  of  the  hypotenuse 
of  a  right  triangle  whose  base  is  12  inches  and  perpen- 
dicular 10  inches,  carrying   the  work   to   two  places  of 

decimals. 

70.  Find  the  diagonal  of  a  square  whose  side  is  8  feet. 

71.  Can  you  show  from  the  accom- 
panying figure  that  the  square  of  the 
diagonal  of  a  square  is  twice  the 
original  square  ? 

Suggestion.  —  Draw  construction  lines 
DF  and  DE. 

72.  ABC  is  an  isosceles  triangle 
right  angled  at  J5,  and  BD  is  its  alti- 
tude. Which  is  the  greater,  the 
square  of  BD  or  the  triangle  ABC? 
Argue  the  point. 


SQUARES   AND  CUBES,  I79 

73.  How  long  a  rope  is  required  to  reach  from  the  top 
of  a  pole  60  feet  high,  to  a  point  on  the  ground  80  feet 
from  the  center  of  its  base  ? 

74.  The  diameter  of  a  round  cistern  is  12  feet,  and  the 
water  is  5  feet  deep.  Find  the  length  of  the  longest 
stick  pointed  at  both  ends  that  can  touch  at  the  same 
time  the  bottom  of  the  cistern  and  the  surface  of  the 
water. 

Query.  —  Can  you  imagine  the  cistern  and  stick  ? 

75.  Show  that  x^  =  a^  -{-  b^  when  a  and  b  are  lines 
respectively  27  inches  and  36  inches,  including  a  right 
angle,  and  a;  is  a  line  45  inches  joining  their  extremities. 

76.  How  can  you  find  the  hypotenuse  of  a  right 
triangle  when  the  base  and  perpendicular  are  given  ? 

77.  Show  that  a?^  =  a^  —  b\  when  a,  the  hypotenuse  of 
a  right  triangle,  is  39  inches,  h,  the  base,  is  15  inches,  and 
x^  the  perpendicular,  is  36  inches. 

78.  Given  the  hypotenuse  and  base  of  a  right  triangle, 
how  can  you  find  the  perpendicular  ? 

79.  Hypotenuse  55  feet,  base  44  feet,  perpendicular  =  ? 

80.  Hypotenuse  39  feet,  base  36  feet,  perpendicular  =  ? 

81.  A  ladder  26  feet  long  is  leaning  against  a  house, 
and  its  base  is  10  feet  from  the  base  of  the  house.  How 
high  does  it  reach  ? 

82.  One  of  the  equal  sides  of  an  isosceles  triangle  is 
25  inches,  and  its  base  is  14  inches.     Find  its  altitude. 

83.  The  perimeter  of  an  isosceles  triangle  is  144  inches, 
and  its  base  is  40  inches.     Find  its  altitude  and  area. 


180  SQUARES  AND  CUBES, 

84.  The  perimeter  of  an  isosceles  triangle  is  50  feet. 
Each  of  the  equal  sides  is  1  foot  longer  than  the  base. 
Find  its  altitude  and  area. 

85.  Given  the  hypotenuse  and  perpendicular  of  a  right 
triangle  ;  how  do  you  tind  the  base  ? 

86.  Hypotenuse  75  feet,  ijer|)endicular  72  feet,  base  =? 

87.  Hypotenuse  78  feet,  perpendicular  72  feet,  base  =? 

88.  What  is  the  perimeter  of  a  right  triangle  whose 
perpendicular  is  24  centimeters  and  whose  hypotenuse  is 
25  centimeters  ? 

89.  One  side  of  an  isosceles  triangle  is  13  feet,  the  alti- 
tude is  12  feet.     Find  the  base  and  area  of  the  triangle. 

90.  The  sum  of  the  equal  sides  of  an  isosceles  triangle  is 
30  feet,  and  the  altitude  is  12  feet.    Find  the  base  and  area. 

91.  How  far  away  from  the  base  of  a  house  must  a 
20-foot  ladder  be  placed  that  its  top  may  reach  a  window 
16  feet  from  the  ground  ? 

92.  The  distance  from  the  center  of  a  circle  whose 
radius  is  10  inches  to  the  middle  point  of  a  certain  chord 
is  8  inches.     How  long  is  the  chord  ? 

93.  Find  the  perimeter  of  a  rectangle  one  side  of  which 
is  30  meters,  and  the  diagonal  of  which  is  50  meters. 

94.  A  house  is  30  feet  wide,  and  the  perpendicular  dis- 
tance from  the  level  of  the  eaves  to  the  comb  uf  the  roof  is  8 
feet.     Find  the  length  of  the  raftei-s. 

95.  AB,  18  inches,  is  tangent  at  its 
middle  point,  (7,  to  the  circle  whose 
diameter  is  12  inches.  Find  the  length 
of  JEB, 


SQUARES  AND  CUBES.  \%\ 

96.  One  of  the  legs  of  an  isosceles  right  triangle  is  8 
feet.  Find  the  base.  Prove  that  the  altitude  is  one  half 
the  base,  using  the  figure  in  Ex.  72. 

97.  One  of  the  legs  of  ^an  isosceles  right  triangle  is  10 
feet.     Find  its  altitude  and  area. 

98.  How  many  times  is  an  isosceles  right  triangle 
contained  in  the  square  of  its  base? 

99.  Find  the  altitude  of  an  equilateral  triangle  each 
of  whose  sides  is  10  feet,  carrying  it  to  two  places  of 
decimals. 

100.  Find  the  approximate  altitude  and  area  of  an 
equilateral  triangle  whose  side  is  8  feet. 

101.  Find  the  area  of  a  rhombus  whose  side  is  9  feet 
and  whose  shorter  diagonal  equals  a  side. 

102.  Can  you  show  that  the  diagonals  of  a  rhombus 
bisect  each  other  at  right  angles? 

Suggestion.  —  See  Geom.  Prin.  25. 

103.  The  diagonals  of  a  rhombus  are  12  inches  and  16 
inches.  Find  a  side  of  the  rhombus.  What  is  the  area 
of  the  rhombus? 

104.  The  sum  of  the  diagonals  of  a  rhombus  is  3-1 
inches,  and  the  long  diagonal  is  14  inches  longer  than  the 
short  diagonal.  Find  the  diagonals,  perimeter,  and  area 
of  the  rhombus. 

105.  The  long  diagonal  of  a  rhombus  whose  side  is  26 
centimeters  is  48  centimeters.     Find  the  short  diagonal. 

106.  Find  the  area  of  an  equilateral  triangle  each  side 
of  which  is  12  inches. 


Ig2  SQUARES  AND  CUBES. 

107.  Place  six  equilateral  triangles  each  of  whose  sides 
is  12  centimeters  around  a  common  point,  and  find  the 
approximate  area  of  the  hexagon  thus  formed. 

108.  Include  an  angle  of  108°  by  two  lines  each  6 
inches.  At  the  extremity  of  one  of  the  lines  make  an- 
other angle  of  108°  on  the  same  side  of  the  line.  Continue 
this  process,  using  6-inch  lines  until  a  surface  is  inclosed, 
and  write  its  name. 

109.  Connect  each  vertex  of  a  regular  pentagon  with 
its  center.  How  many  and  what  kind  of  triangles  are 
thus  formed  ?  Join  the  middle  point  of  one  of  the  trian- 
gles with  the  center  of  the  i)entagon,  and  write  the  name 
of  the  joining  line  upon  it. 

110.  Draw  a  regular  octagon  whose  sides  are  each  4 
inches  by  the  method  given  in  Ex.  108,  making  angles 
of  135°.  Measure  its  apothem,  and  find  its  approximate 
area. 

111.  Draw  by  the  same  method  a  regular  dodecagon 
whose  sides  are  each  4  inches.    Measure,  and  compute  area. 

112.  How  long  will  it  take  a  spider  traveling  3  inches 
a  second  to  cross  diagonally  a  floor  28  feet  long  and  21 
feet  wide  ? 

113.  Join  the  middle  points  of  two  cidjacent  sides  of  a 
square,  and  cut  along  the  line  thus  drawn.  What  frac- 
tional part  of  the  square  has  been  cut  off?     Prove. 

114.  Given  the  square  ABCD,  whose  area  is  64  square 
inches;  JST,   G,  E^  and  F  are  the  middle 
points   of    their   respective   lines.      Find 
the     area     of      the     irregular     hexagon 
QBEFDH,     Of  the  trapezoid  DHGB. 

115.  In  the  same  figure  find  the  length 


SQUARES  AND  CUBES.  183 

of  diagonal  DB.     Of  line  ffCr.     Find  the  perimeter  of 
trapezoid  DHGB,  and  of  hexagon  GBEFDH. 

116.  Find  the  length  of  a  chord  of  90°  in  a  circle  whose 
radius  is  6  inches. 

117.  Find  the  perimeter  of  a  segment  whose  arc  is  90° 
in  a  circle  whose  radius  is  7  inches. 

118.  Cut  a  right  triangle  out  of  pasteboard  and,  keep- 
ing the  perpendicular  upright,  revolve  the  triangle.  What 
geometric  solid  is  formed  by  its  path?  What  surface  is 
formed  by  the  path  of  the  hypotenuse  ? 

119.  The  altitude  of  a  cone  is  the  perpendicular  dis- 
tance from  the  vertex  to  the  base.  The  slant  height  of  a 
cone  is  the  distance  from  the  vertex  to  the  circumference 
of  the  base.  If  the  perpendicular  of  a  cone  is  12  inches 
and  the  radius  of  the  base  is  5  inches,  what  is  its  slant 
height?     What  is  the  area  of  its  base? 

120.  What  is  the  slant  height  of  a  cone  whose  altitude 
is  24  inches,  and  whose  diameter  is  14  inches? 

121.  What  is  the  length  of  the  longest  line  that  can  be 
drawn  on  a  blackboard  12  feet  long  and  5  feet  wide? 

122.  How  long  is  the  longest  stick  that  can  be  carried 
diagonally  through  a  doorway  8  feet  high  and  6  feet  wide? 

Query.  —  Must  the  ends  of  the  stick  be  sharp  pointed? 

123.  What  is  the  diameter  of  the  largest  circular  saw 
that  can  be  carried  through  a  gateway  12  feet  wide  and 
9  feet  high? 

124.  The  perpendicular  of  a  right  triangle  is  9  inches 
longer  than  the  base.  The  perimeter  is  108  inches,  and 
the  hypotenuse  is  45  inches.  Find  the  altitude,  base,  and 
area  of  the  triangle. 


184  SQUARES  AND  CUBES. 

125.  A  regular  hexagon  whose  side  is  5  inches  is  drawn 
within  a  circle  whose  diameter  is  17J  inches.  What  is  the 
area  of  the  surface  between  the  perimeter  of  the  hexagon 
and  the  circumference  of  the  circle  ?     Draw  diagram. 

126.  A  regular  hexagon  is  inscribed  in  a  circle  whose 
radius  is  28  inches.  How  many  square  inches  are  in  the 
sum  of  all  the  segments  formed  by  the  sides  of  the  hexagon  ? 

127.  An  equilateral  triangle  is  inscribed  in  the  same 
circle.  Find  the  area  of  the  segment*  cut  ofif  by  the  sides 
of  the  triangle. 

Query.  —  An  inscribed  equilateral  triangle  is  what  fractional  part 
of  a  regular  hexagon  inscribed  in  the  same  circle? 

128.  Find  the  area  of  a  segment  whose  arc  is  120®  in  a 
circle  whose  radius  is  63  inclies. 

129.  Place  two  squares,  one  8  inches,  the  other  6  inches, 
in  dimensions,  so  that  a  right  angle  is  formed  by  their 
sides.  How  would  you  find  the  side  of  the  square  which 
is  equivalent  to  their  sum  ? 

SuooESTiox.  —  See  Geom.  Prin.  54. 

180.  Find  the  side  of  a  square  whicli  is  equivalent  to 
the  sum  of  a  square  8  inches  and  one  6  inches  in  dimen- 
sions. 

181.  Find  the  perimeter  of  a  square  whose  area  is  equal 
to  that  of  two  squares,  one  9  inches,  the  other  12  inches 
in  dimensions. 

182.  Define  similar  polygons. 

183.  Draw  a  right  triangle  whose  base  is  5  centimeters 
and  perpendicular  8  centimeters.  Draw  another  whose 
base  is  to  the  base  of  the  first  triangle  as  2:1,  and  whose 
perpendicular  is  in  the  same  ratio  to  the  perpendicular  of 
the  first  triangle.     Find  the  ratio  of  their  areas. 


SQUARES  AND   CUBES.  185 

134.  Draw  right  triangles  whose  homologous  sides  are 
in  the  ratio  of  3  to  1,  and  show  by  numbers  that  their 
areas  are  to  each  other  as  9  to  1. 

135.  You  have  shown  that  the  areas  of  similar  right 
triangles  are  to  each  other  as  the  squares  of  their  homolo- 
gous sides.  In  the  same  way  show  the  ratio  of  the  areas 
of  similar  rectangles. 

136.  Are  circles  similar  figures?  Show  that  the  areas 
of  circles  are  to  each  other  as  the  squares  of  their  homolo- 
gous lines. 

Note.  —  Demonstrative  geometry  will  show  you  the  truth  of  the 
following  principle : 

Principle  55.  —  Ths  areas  of  similar  polygons  are 
to  each  other  as  the  squares  of  their  homologous  sides. 

137.  The  area  of  a  heptagon,  one  of  whose  sides  is  6 
inches  long,  is  100  square  inches.  Find  the  area  of  a 
similar  heptagon  whose  corresponding  side  is  9  inches 
long. 

138.  Find  the  area  of  a  pentagon  whose  shortest  side 
is  4  centimeters  when  the  area  of  a  similar  pentagon 
whose  shortest  side  is  5  centimeters  is  known  to  be  50 
square  centimeters. 

139.  A  field,  one  side  of  which  is  20  rods,  contains  5 
acres.  How  many  acres  are  there  in  a  similar  field,  whose 
corresponding  side  is  15  rods  ? 

140.  Mary  dressed  two  dolls  which  were  exactly 
similar,  but  of  different  sizes.  The  hem  of  the  apron 
of  the  larger  doll  was  10  inches  long  and  its  area 
was  48  square  inches.  The  hem  of  the  apron  of  the 
smaller  doll  was  5  inches  long.  What  was  the  area  of 
its  apron? 


186  SQUARES  AND   CUBES. 

141.  The  area  of  a  polygon  whose  longest  side  is  12 
inches  is  36  square  inches.  Find  the  longest  side  of  a 
similar  polygon  whose  area  is  64  square  inches. 

Suggestion.  —  Let  x  =  the  required  side. 

142.  Complete  the  table  of  the  cubes  of  the  integers 
from  1  to  25  inclusive.     1^=1,  2^=8,  etc. 

Derive  a  table  of  the  cubes  of  10,  20,  etc.,  from  the 
preceding  table. 

143.  Build  a  cube  of  a  line  4  inches  long. 

Note.  —  A  cube  which  has  a  certain  line  for  each  of  its  edges  is 
said  to  be  the  cube  of  that  line. 

144.  How  many  faces  and  how  many  edges  has  a  cube? 

145.  How  many  1-inch  cubes  make  the  first  layer  of 
a  4-inch  cube  ?     How  many  are  there  in  all  the  layers? 

146.  Give  the  volume  and  the  superficial  area  of  a:*,  z 
being  a  5-inch  line. 

147.  How  many  1-inch  cubes  are  there  in  the  lowest 
layer  of  a  cube  which  is  8  inches  in  dimensions?  How 
many  layers  compose  the  cube  ? 

148.  What  is  the  length  of  one  edge  of  a  cube  which 
contains  2197  cubic  inches  ?     Of  all  its  edges  ? 

149.  What  is  the  area  of  one  face  of  a  cube  which  con- 
tains 5832  cubic  inches  ?     Of  all  its  faces  ? 

150.  What  is  the  volume  of  a  cube  one  face  of  which 
contains  81  square  centimeters  ? 

151.  What  is  the  volume  of  a  cube  whose  faces  contain 
294  square  decimeters  ? 

152.  How  lonor  is  the  side  of  a  cubical  vessel  containing 
343  cubic  centimeters  ? 


SQUARES  AND   CUBES.  Ig7 

153.  The  length,  height,  and  width  of  a  certain  room 
are  equal.  It  contains  729  cubic  feet.  Find  the  distance 
diagonally  across  the  floor  of  the  room. 

154.  Find  the  sum  of  all  the  edges  of  a  cube  which 
contains  125  cubic  centimeters. 

155.  Find  the  area  of  all  the  faces  of  a  cube  which  con- 
tains 1728  cubic  inches. 

156.  Find  the  sum  of  all  the  diagonals  which  can  be 
drawn  on  the  faces  of  a  cube  containing  216  cubic  inches. 

157.  Find  the  sum  of  the  perimeters  of  all  the  faces  of 
a  cube  which  contains  512  cubic  inches. 

158.  Find  the  area  of  all  the  squares  which  can  be 
inscribed  in  the  faces  of  a  cube  containing  1331  cubic 
inches. 

159.  How  many  cubes,  one  inch  each  way,  can  be  cut 
from  a  cube  4  inches  in  its  dimensions  ? 

160.  How  many  cubes,  each  one  quarter  of  an  inch  in 
its  dimensions,  can  be  cut  from  an  inch  cube  ? 

161.  How  many  inch  cubes  can  be  cut  from  a  3-inch 
cube? 

162.  How  many  2-inch  cubes  can  be  cut  from  a  6-inch 
cube  ? 

163.  How  many  3-inch  cubes  can  be  cut  from  a  6-inch 
cube? 

164.  Find  the  edge  of  the  cube  which  is  the  G.C.M. 
of  two  cubes,  one  containing  27  cubic  inches  and  the 
other  216  cubic  inches.  Find  the  edge  of  the  smallest 
cube  that  will  contain  each  of  them  a  whole  number  of 
times. 


188  SQUARES  AND  CUBES. 

165.  Find  the  edge  of  a  cube  which  is  the  L.C.M.  of 
two  cubes,  one  containing  216  cubic  inches,  the  other 
64  cubic  inches.  Find  the  largest  cube  which  can  be  cut 
from  each  of  them  a  whole  number  of  times. 

166.  How  many  cubic  centimeters  are  there  in  a  liter? 
How  many  cubic  millimeters  are  there  in  a  cubic  centi- 
meter ?  „ 

Query.  —  Can  you  make  a  cubic  millimeter? 

167.  Find  the  sum  of  all  the  edges  of  a  cube  containing 
1728  cubic  inches. 

168.  Find  the  sum  of  all  the  edges  of  7  cubes,  each 
containing  64  cubic  inches. 

169.  Find  the  area  of  all  the  surfaces  of  a  cube  whose 
contents  are  125  cubic  inches. 

170.  Find  the  area  of  the  surface  of  a  box  twice  as 
long  as  wide,  whose  width  equals  its  height,  and  whose 
contents  are  128  cubic  inches. 

Suggestion.  —  Mold  the  shape  in  clay,  or  make  it  in  paat«board, 
or  cut  it  out  of  a  potato  or  turuip,  or,  best  of  all,  form  a  clear  mental 
image  of  it. 

171.  Find  the  area  of  the  surfaces  of  a  box  whose 
width  and  height  are  equal,  length  twice  the  width,  and 
contents  250  cubic  inches. 

172.  Give  the  volume  of  a  parallelopij^ed  5  inches  long, 
4  inches  wide,  and  3  inches  high,  and  the  area  of  its 
sui'faces. 

173.  Find  the  cubic  contents  and  area  of  inside  surfaces 
of  a  box  whose  inside  dimensions  are  8  inches,  6  inches, 
and  4  inches. 


SQUARES  AND  CUBES.  Igg 

174.    Given  x  a  3-inch  line,  z/  a  2-inch  line.  Build  x^ ; 

build  3/3;    build   the    rectangular   solid   3^1/;  build  the 

rectangular  solid  a:/.     Find  tlie  volume  and  facial  area 
of  each. 

1T5.  Given  x  a  2-inch  line,  y  a  1-inch  line.  Build 
a  solid  which  represents  x^-hy^  one  angle  of  the  smaller 
cube  coinciding  with  one  angle  of  the  larger.  Find  the 
area  of  its  surfaces. 

176.  Cut  an  inch  cube  out  of  a  corner  of  a  2-inch  cube, 
and  find  the  volume  of  the  remaining  solid  and  the  area 
of  its  surfaces. 

177.  Find  the  volume  and  area  of  surfaces  of  the  solid 
represented  by  a;3_^3^  when  x  =  'd  Hue  of  4  inches  and  y 
=  a  line  of  3  inches,  y^  being  cut  from  a  corner  of  2^, 

178.  Given  x  a  3-inch  line,  1/  a  2-inch  line.  Build  a 
solid  composed  of  a^,  Sx^t/,  Sxi/^,  and  1/^,  making  it  a.s 
nearly  as  possible  in  the  shape  of  a  cube. 

179.  Show,  by  using  blocks,  that  the  cube  of  the  sum 
of  two  lines  equals  the  cube  of  the  first,  plus  3  times  the 
square  of  the  first  multiplied  by  the  second,  plus  3  times 
the  square  of  the  second  multiplied  by  the  first,  plus  the 
cube  of  the  second. 

180.  Find  a^,  3  x^i/,  3  xt/^,  and  ^,  when  2;  =  a  8-inch  line 
and  1/  an  inch  line  and  combine  them  into  a  cube. 

181.  Find  the  area  of  the  surfaces  of  the  irregular 
solid  formed  by  placing  the  cube  of  a  line  of  4  inches 
upon  the  cube  of  a  line  of  5  inches,  in  such  a  way  that  the 
middle  point  of  the  lower  base  of  the  smaller  cube  coin- 
cides with  the  middle  point  of  the  upper  base  of  the 
larger,  and  their  edges  are  parallel. 


190  SQUARES  AND   CUBES. 

182.  What  is  the  ratio  of  a  1-inch  cube  to  a  3-inch 
cube  ?     Of  a  2-inch^  cube  to  a  5-inch  cube  ? 

183.  Can  you  show  that  the  volumes  of  two  cube**  are 
in  the  ratio  of  the  cubes  of  their  edges  ?  Are  they  simihir 
solids  ? 

Note.  —  Demonstrative  geometry  will  show  you  the  truth  of  three 
closely  related  principles  which  will  be  very  useful  to  you. 

Principle  56.  —  On  similar  solids  homologous  lines 
are  proportional  to  other  hom^lo^ous  lines. 

Principle  57.  —  On  similar  solids  homologoics  sur- 
faces are  proportional  to  the  squares  of  homologous 
lines. 

Principle  58.  —  The  volum^es  of  sim^Uar  solids  are 
proportional  to  the  cubes  of  homologous  lines. 

184.  A  manufacturer  makes  stoves  of  similar  patterns 
but  different  sizes.  Size  No.  4  is  3  feet  in  height;  size 
No.  18  is  6  feet  in  height.  What  is  the  width  of  the  door 
of  the  larger  stove  if  the  corre8iK)nding  door  of  tlie 
smaller  is  6  inches  wide  ?  A  brass  rod  on  the  larger  one 
is  4  feet  2  inches  long.  How  long  is  the  corresponding 
rod  on  the  smaller  one  ?  The  legs  of  the  larger  stove 
raise  it  10  inches  from  the  floor.  How  far  is  the  smaller 
from  the  floor  ? 

185.  The  area  of  a  door  of  No.  18  is  72  square  inches. 
How  many  square  inches  of  iron  plate  are  used  in  making 
the  corresponding  door  of  No.  4  ?  The  hearth  of  No.  4 
contains  20  square  inches.  How  many  square  inches  are 
there  in  the  hearth  of  No.  18  ?  No.  4  requires  36  square 
inches  of  mica.  How  many  square  inches  of  mica  will 
No.  18  require  ?     No.  18  has  80  square  inches  of  nickel 


SQUARES  AND  CUBES.  191 

finish.     How  many  square  inches  of  nickel  finish  are  used 
upon  No.  4  ? 

186.  The  cubic  contents  of  No.  18  are  how  many  times 
the  cubic  contents  of  No.  4?  If  75  pounds  of  iron  are 
used  in  making  No.  4,  how  many  are  required  for  No.  18  ? 

187.  Mary's  doll,  which  is  10  inches  long,  is  exactly 
similar  to  Anna's  doll,  which  is  30  inches  long,  and  the 
dolls  are  dressed  alike.  If  a  hand  of  Mary's  doll  is  1 
centimeter  long,  what  is  the  length  of  a  hand  of  Anna's 
doll  ?  If  the  belt  of  Anna's  doll  is  12  inches  long,  how 
long  is  the  belt  of  Mary's  doll  ?  If  the  hem  of  the  dress 
of  Anna's  doll  is  33  inches  long,  how  long  is  the  hem  of 
the  dress  of  Mary's  doll  ? 

188.  If  it  took  half  a  yard  of  pink  satin  to  make 
the  dress  for  Mary's  doll,  how  many  yards  will  it  take 
for  the  dress  of  Anna's  ?  If  it  took  one  square  yard  of 
wliite  plush  to  make  a  wrap  for  Anna's  doll,  how  many 
square  yards  are  required  for  a  similar  garment  for  Mary's 
doll? 

189.  If  Mary's  doll  weighs  5  ounces,  how  many  pounds 
and  ounces  does  Anna's  weigh  ? 

190.  How  many  iron  balls  2  inches  in  diameter  will 
equal  in  weight  an  iron  ball  10  inches  in  diameter? 

191.  A  vase  8  inches  high  holds  72  cubic  inches. 
How  many  cubic  inches  are  there  in  a  vase  of  the  same 
shape  which  is  4  inches  high  ? 


CHAPTER  XV. 


CUMULATIVE  REVIEW 

1.  BA  is  perpendicular  to  AC\  DE 
is  parallel  to  AC,  £A  =  2\  centi- 
meters; -4(7=20  centimeters;  find  BC, 
BD=\A:  centimeters;  find  DA^  EC, 
and  DE.  Quote  the  geometric  princi- 
ples used  in  solving  this  problem. 

2.  The  radius  of  a  circle  is  10  inches.  Find  the  dis- 
tance from  the  center  of  the  circle  to  the  middle  point  of 
a  chord  of  12  inches. 


3.  The  chord  AB  is  18  inches;  radius,  15 
inches.  Find  the  length  of  AC,  the  chord  of 
half  the  arc  ^(Tfi. 

Suggestion.  —  Find  Z)0,  then  DC, 

4.  Make  and  solve  an  original  problem  whose  solution 
depends  upon  the  fact  that  a  radius  perpendicular  to  a 
chord  bisects  it. 

5.  Express  in  its  lowest  terms  the  fractional  part  which 
an  arc  of  ^%°  is  of  the  circumference. 

6.  A  certain  circumference  is  divided  into  three  arcs. 
The  first  is  50°  more  tlian  the  second,  and  the  second  50® 
more  than  the  third.     Find  each. 

192 


CUMULATIVE  HE  VIEW.  I93 

7.  How  long  is  each  arc  given  in  Ex.  6  if  the  diame- 
ter of  the  circle  is  21  feet  ? 

8.  How  many  degrees  are  there  iu  an  arc  of  a  circle 
if  the  remaining  arc  is  5  times  as  large  ?  How  long  is 
it  if  the  radius  is  15  centimeters  ? 

9.  Find  the  perimeter  of  a  sector  of  90°  in  a  circle 
whose  radius  is  17^  inches. 

10.  Find  the  perimeter  of  a  sector  of  SO**  of  a  circle 
whose  circumference  is  16 J  inches. 

11.  Find  the  perimeter  of  a  segment  of  90°  in  a  circle 
Avhose  radius  is  28  centimeters. 

12.  Find  the  perimeter  of  a  segment  of  60°  in  a  circle 
whose  radius  is  SS^  inches. 

13.  Find  the  difference  between  the  perimeter  of  a 
segment  of  180°  and  that  of  a  sector  of  180°  in  a  circle 
whose  radius  is  52J  millimeters. 

14.  The  complement  of  a  certain  angle  is  20°  more 
than  twice  the  angle.  How  many  degrees  are  there  in 
the  supplement  of  the  angle  ? 

15.  Find  the  angle  which  is  the  G.  C.  M.  of  the  three 
angles  formed  at  the  center  of  a  circle,  the  arc  of  one . 
being  10°  greater  than  one  of  its  adjoining  arcs,  and  10° 
less  than  the  other. 

16.  How  many  degrees  are  there  in  the  arc  of  the 
sector  which  is  the  G.  C.  M.  of  two  sectors  of  the  same 
circle,  whose  arcs  are  80°  and  60°  ? 

17.  How  many  degrees  are  there  in  the  arc  of  the 
sector  which  is  the  L.  C.  M.  of  two  sectors  of  the  same 
circle,  the  arcs  of  the  sectors  being  30°  and  40°  ? 

HORN.    0£0M.  —  13 


194  CUMULATIVE  REVIEW. 

18.  When  is  a  segment  also  a  sector  ? 

19.  Draw  a  circle,  and  inscribe  in  it  two  right  triangles 
in  such  a  way  that  they  shall  not  overlap  upon  each  other. 

20.  Two  arcs,  one  of  which  is  30°  more  than  4  times  the 
other,  compose  a  circumference.  How  many  degrees  are 
there  in  each  of  the  angles  which  they  measure?  a 

21.  AD  is  a  diameter.  Arc  DC  =  50°. 
Arc  BB  =  80°.  Find  each  angle  of  the  tri- 
angle ABO. 

22.  How  long  is  the  arc  intercepted  by 
an  inscribed  angle  of  30°  in  a  circle  whose  radius  is  12 
inches  ? 

23  Which  is  greater,  the  angle  ABC 
formed  by  two  secants  intercepting  arc  AC^ 
or  the  inscribed  angle  ADC^  whose  sides  in- 
tercept the  same  arc  ?     Give  reasons. 

24.  How  many  times  will  a  wheel  6  feet 
6  inches  in  diameter  revolve  in  traveling  44 
feet? 

25.  Find  the  distance  from  the  center  of  a  circle  whose 
radius  is  13  inches  to  the  middle  of  a  chord  24  inches 
long. 

2G.  With  a  radius  of  6  inches  draw  a 
circle  and  two  radii  perpendicular  to  each 
other.  Bisect  by  a  radius  the  angle  formed 
by  them.  Draw  a  tangent  at  the  extremity 
of  the  radius.  Prolong  the  radii  which 
form  the  angle  until  they  meet  the  tan- 
gent.    What  is  the  area  of  the  triangle  thus  formed  ? 


CUMULATIVE  REVIEW.  195 

27.  The  perimeter  of  an  isosceles  triangle  is  72  inches. 
Each  of  the  legs  is  6  inches  longer  than  the  base.  Find 
each  side.     Find  the  altitude  and  area. 

28.  The  altitude  of  an  inscribed  equilateral  triangle  is 
what  fractional  part  of  the  diameter  of  the  circl©  ? 

29.  The  side  AB  of  the  scalene  triangle  ABC  is  7 
inches  longer  than  the  side  BC,  and  BO  is  S  inches  longer 
than^C     The  perimeter  is  25  inches.     Find  each  side. 

30.  The  longest  side  of  a  scalene  triangle  is  5  inches 
longer  than  the  next  in  length,  and  the  next  in  length  is 
twice  the  length  of  the  shorter  side.  Find  the  length  of 
each  side  if  the  perimeter  is  55  inches. 

31.  The  sum  of  the  legs  of  an  isosceles  triangle  is  12 
meters  more  than  the  base,  and  the  perimeter  is  48  meters. 
Find  each  side.     Find  the  altitude  and  area. 

32.  Find  the  angles  of  an  isosceles  triangle  in  which 
the  vertical  angle  is  J  of  the  sum  of  the  base  angle. 

33.  A  triangle  whose  base  is  12  centimeters  and  alti- 
tude 3  millimeters  is  what  fractional  part  of  one  whose 
base  is  16|  centimeters  and  altitude  4J  millimeters  ? 

34.  Draw  an  equilateral  triangle  whose  side  is  3  inches, 
and  find  the  ratio  of  its  surface  to  that  of  an  equilateral 
triangle  whose  side  is  one  inch. 

35.  Find  the  ratio  of  the  surfaces  of  two  equikteral 
triangles,  the  ratio  of  whose  sides  is  4  :  3. 

36.  AB  is  36  inches.  CD,  perpendicular 
to  it,  is  9  inches  ;  BU,  the  perpendicular  to 
the  line  AC,  is  20  inches.     Find  AC. 

37.  The  area  of  an  isosceles  triangle  is  1452  square  feet, 
its  altitude  is  44  feet.     Find  its  perimeter. 


„^. 


196  CUMULATIVE  REVIEW. 

38.  The  sum  of  the  equal  sides  of  an  isosceles  triangle 
and  its  altitude  equals  49  inches.  The  difference  between 
one  of  them  and  the  altitude  equals  2  inches.  The  base 
is  16  inches.     Find  the  area  and  perimeter. 

Queries.  —  Shall  a:  =  the  altitude  or  one  of  the  legs  of  the  tri- 
angle ?  Can  you  draw  an  isosceles  triangle  having  its  altitude  equal 
to  or  greater  than  a  leg  ? 

39.  Draw  the  line  AB^  and  a  line  every  point  of  which 
is  the  same  distance  from  A  as  from  B. 

Quote  the  geometric  principle  which  guides  you. 

40.  Find  the  altitude  and  area  of  an  isosceles  triangle 
whose  base  is  40  millimeters,  and  whose  equal  sides  are 
each  29  millimeters. 

41.  Construct  equilateral  triangles  upon  each  side  of  a 
square  whose  side  is  8  inches,  and  find  the  area  of  the 
starred  figure  thus  formed. 

42.  The  length  of  a  rectangle  is  18  millimeters,  and  its 
width  is  J  of  the  length.  If  its  width  is  reduced  3  milli- 
meters, how  much  must  be  added  to  the  length  that  the 
area  may  be  the  same  ? 

43.  The  width  of  a  rectangle  is  12  feet,  and  its  length 
1^  times  as  much.  If  the  length  is  decreased  4  feet  and 
the  width  increased  4  feet,  what  figure  is  formed?  By 
how  much  is  the  original  rectangle  increased  ? 

44.  Draw  perpendiculars  to  each  extremity  of  both 
diagonals  of  a  square,  and  prolong  them  until  a  surface 
is  inclosed.  What  figure  is  thus  formed,  and  what  ratio 
does  it  bear  to  the  given  square  ? 

45.  Find  the  number  of  square  centimeters  in  the 
surface  of  a  parallelopiped  4  decimeters  long,  3  decime- 


CUMULATIVE  REVIEW.  I97 

ters  wide,  and  2  decimeters  high.     How  many  liters  are 
there  in  the  parallelepiped  ? 

46.  Draw  a  line  5  inches  and  another  6  inches,  and 
show  that  the  square  of  the  first  plus  twice  the  rectangle 
of  their  product,  plus  the  square  of  the  second,  will  give 
a  square  11  inches  in  its  dimensions. 

47.  Cut  out  two  figures,  one  the  rectangle  of  the  sum 
and  difference  of  two  lines,  the  other  the  difference  of 
their  squares,  and  cut  and  fit  one  upon  the  other. 

48.  Find  the  area  of  a  regular  hexagon,  one  side  of 
which  is  12  centimeters. 

49.  The  perimeter  of  a  rhomboid  is  24  centimeters. 
The  difference  between  two  adjacent  sides  is  2  centime- 
ters.    Find  each  side. 

50.  Which  is  greater,  a  square  whose  sides  are  each 
5  inches,  or  a  rhomboid  whose  base  and  altitude  are  each 
5  inches  ?     Which  has  the  greater  perimeter  ? 

51.  Two  horizontal  parallels  8  inches  apart  are  crossed 
by  two  vertical  parallels  also  8  inches  apart.  What  is  the 
inclosed  surface  and  what  is  its  area  ?  When  the  vertical 
parallels  are  inclined  so  as  to  cross  at  an  angle  of  ()0^  the 
intersections  still  being  8  inches  apart,  what  is  the  name 
of  the  figure  thus  formed,  and  what  is  its  area  ? 

52.  The  side  of  a  square  is  1  foot  longer  than  that 
of  another  square,  and  the  sum  of  their  perimeters  is 
84  feet.     Find  the  sum  of  their  squares. 

53.  AE  the  altitude  =  10  centimeters ; 
DE  =  3  centimeters  ;  EC  =  8  centime- 
ters ;  find  the  area  of  rhomboid  A  BCD. 


198 


CUMULATIVE  REVIEW. 


54.  AD  =  \  oi  AB.     BE  is  parallel  to  BQ.    ^ 
The  perimeter  of  triangle  ABC  is  30  inches. 
Find  the  perimeter  of  triangle  ABE^  and  quote 
the  appropriate  geometric  principles. 

55.  A  long  side  of  a  rhomboid  is  twice  a  short  side, 
and  the  perimeter  is  54  inches.  Find  the  perimeter  of 
a  similar  rhomboid  whose  short  sides  are  each  6  inches. 

56.  ^^ira  is  60  inches.  AQ^n^BE 
are  equal,  and  each  is  twice  AB  and  6 
inches  less  than  QH,  The  trapezoids 
are  similar.  CB  is  4 J  inches.  Find  each 
side  of  each  trapezoid.  o^ 

57.  The  trapezium  ABCB  has  the  sides  BO 
and  CB  equal.  AB  is  8  inches  longer  than 
BC,  and  AB  is  12  inches  longer  than  CB, 
The  perimeter  is  56  inches.  Find  the  line 
which  is  the  greatest  common  measure  of 
the  sides. 

58.  The  perimeters  of  two  similar  trapeziums  are  re- 
spectively 27  inches  and  36  inches.  If  the  shortest  side 
of  the  smaller  is  6  inches,  what  is  the  shortest  side  of  the 

other  ? 

59.  What  is  the  area  of  the  largest  circle  that  can  be 
cut  from  a  piece  of  paper  which  is  14  inches  square  ? 

60.  What  is  the  area  of  the  circular  ring  bounded  by 
the  circumferences  of  two  concentric  circles,  the  diameter 
of  one  being  20  inches  and  of  the  other  10  inches  ? 

61.  A  circular  plat  40  feet  in  diameter  has  a  walk 
around  the  outside  which  is  4  feet  wide.  Find  the  area 
of  the  walk. 


CUMULATIVE  REVIEW.  199 

62.  The  circumference  of  circle  A  is  3  times  that  of 
circle  B.  A  chord  of  60°  on  circle  ^  is  5  centimeters. 
How  long  is  a  chord  of  60°  on  circle  B  ? 

63.  Draw  two  parallels  and  a  transversal  so  that  an 
exterior  angle  shall  be  10°  more  than  its  adjacent  interior 
angle,  and  mark  the  number  of  degrees  in  each  of  the 
eight  angles  formed. 

64.  One  angle  of  a  rhomboid  is  70°.  Find  the  ratio 
of  its  adjacent  exterior  angle  to  the  ratio  of  its  consecu- 
tive angle. 

Qb.  How  do  you  find  the  number  of  degrees  in  the 
angles  of  a  polygon  ? 

QQ.  How  many  degrees  are  there  in  each  angle  of  a 
regular  polygon  of  14  sides? 

67.  How  many  degrees  are  there  in  each  exterior  angle 
of  an  octagon  ?  Of  a  pentagon  ?  Of  a  decagon  ?  Of  a 
dodecagon  ? 

68.  What  is  the  ratio  of  the  sum  of  the  exterior  angles 
of  a  pentagon  to  the  sum  of  the  exterior  angles  of  an 
octagon  ? 

69.  Circumscribe  a  circle  whose  radius  is  6  inches 
around  a  regular  hexagon.  The  perimeter  of  the  hexa- 
gon has  what  ratio  to  the  circumference  ? 

70.  A  and  B  start  from  the  same  point.  A  travels 
north  at  the  rate  of  8  miles  per  hour,  B  travels  west  at 
the  rate  of  six  miles  per  hour.  How  far  apart  are  they  at 
the  end  of  7  hours? 

71.  How  many  times  is  a  square  whose  side  is  3  inches 
contained  in  a  square  whose  side  is  5  times  as  long? 


200  CUMULATIVE  REVIEW. 

72.  How  many  times  is  a  cube  whose  edge  is  2  inches 
contained  in  a  cube  whose  edge  is  5  times  as  long? 

73.  Name  a  plane  figure  that  has  no  angles. 

74.  Hold  a  card  upright  so  that  one  of  its  edges  is  on 
your  desk,  and  keeping  it  in  the  same  position  move  it 
from  one  side  of  the  desk  to  the  other.  The  path  of  the 
upper  edge  is  what  ?  The  path  of  the  card  forms  what 
geometric  solid  ? 

75.  If  the  card  moved  from  one  side  of  the  desk  to  the 
other  were  8  inches  by  5  inches,  the  desk  15  inches  wide, 
and  one  of  the  shorter  edges  were  on  the  desk,  what  would 
be  the  area  of  the  surface  generat<;d  by  the  upper  edge  ? 
What  would  be  the  area  of  the  surfaces  of  the  parallelo- 
piped  generated  by  the  card  ? 

76.  How  many  spheres  can  there  be  which  have  a  com- 
mon center? 

77.  Think  of  two  spheres  of  different  size  having  a 
common  center.     Have  they  parallel  surfaces  ? 

78.  A,  B,  (7,  D,  Ey  and  F  are  points  at 
equal  distances  on  the  circumference  of  the 
circle  whose  radius  is  lOJ  inches,  and  whose 
center  is  G.  By  how  much  does  the  sum  of 
the  perimeters  of  the  triangles  exceed  the 
circumference  of  the  circle  ?  By  how  much 
does  the  sum  of  the  perimeters  of  the  sectors  between  the 
triangles  exceed  the  sum  of  the  perimeters  of  the  triangles  ? 

79.  Find  the  edge  of  a  cube  the  sum  of  whose  faces  is 
150  square  centimeters. 

80.  Find  the  volume  of  a  cube  the  sum  of  whose  edges 
is  72  decimeters.  • 


CUMULATIVE  REVIEW.  201 

81.  What  line  contains  all  the  points  in  a  plane  surface 
which  are  30  inches  distant  from  the  center  of  a  circle 
whose  diameter  is  40  inches.     Illustrate. 

82.  There  are  two  similar  monuments,  of  which  the 
smaller  is  3  feet  high,  the  larger  27  feet  higli.  An  etlge 
of  the  base  of  the  smaller  is  4  feet ;  find  tlie  correHponding 
edge  of  the  larger.  The  area  of  the  top  of  the  larger  is 
324  square  feet ;  find  the  area  of  the  top  of  the  smaller. 
The  weight  of  the  smaller  is  200  pounds ;  find  the  weight 
of  the  larger. 

83.  There  are  two  haystacks  of  similar  shape.  One  is 
10  feet  high  and  contains  4  tons.  How  many  tons  are 
there  in  the  other,  which  is  20  feet  high  ? 

84.  Of  two  cellars  of  similar  shape,  the  length  of  the 
larger,  28  feet,  is  twice  that  of  the  smaller.  The  width  of 
the  larger  is  24  feet,  and  its  depth  is  10  feet.  Find  the 
cubic  contents  of  each.  What  is  the  ratio  of  their  con- 
tents?    Of  their  surfaces? 

85.  Of  two  books  of  similar  shape,  one  is  twice  as  thick 
as  the  other.     If  the  smaller  book  weighs  3  ounces,  how 

much  does  the  larger  one  weigh? 


^      OF  THF 


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on  receipt  0/  the  price y  by  the  Publisher t: 

AMERICAN  BOOK  COMPANY 

NEW  YORK  aNCI^4NATl  CHICAGO 

BOSTON     -    ATLANTA    •    POKTIANIX  ORE. 
(99) 


UNIVERSITY   OK  CAIJFOHN'I A    MBRAUY 


THIS  BOOK  IS  DUE  ON  THE  LAST  DATE 
STAMPED  BELOW 


MAY   4   1915 


XiS^ 


30 


MAH   ^  1^2\ 


^  23  1932 


APR   Hie  »s«l 


AUG  S    192J 


Jot  r  1929 

SEP  21  1931 

30m  6,14 

YB  Gjic; 


■Hj-j  ' 


